NUMERICAL STUDY OF HYPERSONICREACTING BOUNDARY LAYER TRANSITION ON CONES
Heath B. JohnsonTrevor G. Seipp
Graham V. CandlerDepartment of Aerospace Engineering and MechanicsArmy High Performance Computing Research Center
University of MinnesotaMinneapolis, MN 55455
Abstract
Hypersonic gas flow over cones is solved using com-putational fluid dynamics to obtain accurate boundarylayer profiles. A linear stability analysis is performedon the profiles to determine the amplification rates ofnaturally occurring disturbances, and this informationis used to predict the boundary layer transition lo-cation. The effects of freestream total enthalpy andchemical composition on transition location are stud-ied to give a better understanding of recent experi-mental observations, namely, there is an increase intransition Reynolds number with increasing freestreamtotal enthalpy, and this increase is greater for gasseswith lower dissociation energies. The results show thatlinear stability predicts the same trends that were ob-served in the experiments, but it consistently overpre-dicts the transition Reynolds numbers by about a fac-tor of two. The results of numerical experiments arepresented which show the effect of reaction endo- orexothermicity on disturbance amplification rates.
I. Introduction
The transition of boundary layers from laminar toturbulent flow has been the subject of study for manyyears. One of the reasons for interest in this area isthe different properties of the flow in the laminar andturbulent regimes. For example, aerodynamic heatingand skin friction at hypersonic speeds are much greaterfor turbulent flow than for laminar flow. Thus, it isimportant to know whether the flow is laminar or tur-bulent to optimize the design of hypersonic vehiclesand their thermal protection systems. This is espe-cially crucial in reusable vehicles where any increasein the weight of the thermal protection system gives areduction in the payload capacity.
An understanding of the transition process and anaccurate prediction criterion has only recently begunto develop. In the past, empirical relationships basedon edge Mach number and Reynolds number were usedto predict the location of transition on a vehicle. These
relationships were mostly based on reentry flight data.A physical understanding of the parameters affectingthe transition process is essential for developing accu-rate methods for predicting transition analytically.
Stability analysis is one means of studying thephysics of the transition problem. In stability analy-sis, boundary layer transition is assumed to be causedby the amplification of wave-like disturbances that areinherent in the mean flow. In the past, linear stabilitytheory and parabolized stability equations have beenused to evaluate the effects of parameters such as walltemperature, unit Reynolds number, and Mach num-ber on boundary layer stability. Recently, the chemi-cal and thermal state of the gas has been explored as aparameter. Malik and Anderson1 have quantitativelyexamined equilibrium gas, Stuckert and Reed2 havestudied chemical nonequilibrium, and Hudson et al.3
have examined the effects of thermal and chemical non-equilibrium. Several researchers4'5'6 have attemptedto qualitatively imply stability from mean flow pro-files alone.
Germain and Hornung7 and Adam and Hornung,8
using the GALCIT (Graduate Aeronautical Labo-ratories, California Institute of Technology) Free-Piston Shock Tunnel, T5, have produced experimentaldata showing the intriguing result that the transitionReynolds number increases with increasing freestreamtotal enthalpy, ho- Previous to this, experimental in-vestigations of hypersonic transition were limited tocold flows where high Mach numbers were primarilyachieved by lowering the speed of sound, and the ki-netic energy was not large enough to cause dissocia-tion effects to become important.7 While the enthalpyeffect shown in the T5 results is clearly evident for ni-trogen flows, the increase in transition Reynolds num-ber is more pronounced for air and even more so forcarbon dioxide. The hypothesized reason is that theendothermic chemical reactions occurring at higher en-thalpies absorb energy from the instability waves. Re-cent direct numerical simulations ,of reacting isotropicturbulence have shown that endothermic reactions can
dampen energy fluctuations.9 This effect could possi-bly play a role in delaying transition by damping in-cipient instabilities. Other plausible reasons for the in-crease in transition Reynolds number with freestreamenthalpy include boundary layer thickness and non-equilibrium effects in the mean flow. In any case, atransition prediction method which does not take intoaccount the effect of chemical reactions will not giveaccurate stability results. Prom an engineering stand-point, this could lead to excessive design conservatismin hypersonic vehicles.
The idea that the origin of turbulence was inthe instability of laminar flow was first hypothe-sized by Reynolds in the late 1800's. In the 1930's,Tollmien and Schlichting developed a stability the-ory to describe the behavior of wave-like disturbances(Tollmien-Schlichting instability waves) which are thefirst indication of the transition from laminar to turbu-lent flow. Stability theory defines the behavior of thedisturbance waves inherent in the mean laminar flow.If the waves decay, they are stable; if they amplify,they are unstable and presumably lead to transition.
Mack6 showed that there are multiple modes of in-stability for hypersonic flows. The first mode is a vor-ticity disturbance, and the second and higher modesare primarily acoustic modes. The two modes can bedescribed by defining a local relative Mach number, Mas:
M = (u — cr)/a (1)where u is the local velocity parallel to the surface,cr is the disturbance phase velocity, and a is the localspeed of sound. A generalized inflection point, whichrepresents a maximum in angular momentum, is de-fined as the location:
du.(2)
where p is the density and y is the distance normalto the surface. Lees and Lin10 showed for perfect gasflows that if M < I everywhere, the existence of a gen-eralized inflection point is a sufficient condition for theexistence of a first-mode instability. The second modeinstability depends only on the existence of a regionof supersonic mean flow relative to the disturbance
g
phase velocity, M > 1, and there is a multiplicityof unstable modes.6 Two-dimensional modes are themost unstable for second mode instabilities whereasoblique waves are the most unstable for first modeinstabilities. For hypersonic gases, the second modehas been found to be the most unstable for perfectgases,6 for chemical equilibrium,1 and for thermochem-ical nonequilibrium.3
The present work uses computational models toreproduce the shock tube experiments of Germainand Hornung7 and Adam and Hornung8 to exam-ine whether boundary layer stability theory predictsthe observed trends. To that point, a computationalmodel was designed to reproduce the experimentalconditions as closely as possible. A series of tests wererun over a range of enthalpies for nitrogen, air, andcarbon dioxide flows over a 5° half-angle cone. Thesecond-mode disturbance amplification curves werecomputed at twenty locations along the body. Thesedata were then used with the eN method to produce atransition Reynolds number, and the computed tran-sition Reynolds numbers were compared with thosewhich were experimentally measured. In Section II,we describe the numerical method used for the solu-tion of the mean flows. In Section III, we briefly de-scribe the equations of linear stability and their solu-tion. The method used to predict the boundary layertransition locations is described in Section IV. Resultsof the transition calculations and comparison with theexperiments are presented in Section V. In Section VI,we present numerical experiments which demonstratethat the presence of chemical reactions in the bound-ary layer disturbances can be stabilizing. Finally, Sec-tion VII finishes with some concluding remarks.
II. MEAN FLOWA. Governing Equations
The mean flow is described by an extended set ofthe Navier-Stokes equations for a five-species gas mix-ture in thermal and chemical nonequilibrium. A two-temperature model was used where the energy in thetranslational and rotational modes is characterized bythe translational temperature T, and the energy ofthe vibrational modes is characterized by the vibra-tional temperature Tv. Thus, five species equations,two momentum equations, and two energy equationswere solved.
Two different models for reacting gas were used.The first was for reacting air or nitrogen and mod-eled the following three dissociation reactions and twoexchange reactions:
N2 + M02 + M
N2 + 0NO + O
2N + M2O + MN + O +NO + NO2 + N
where M is any collision partner (such as N2, O2,NO, N, or 0). The equilibrium constants, transport
properties, and translational-vibrational energy ex-change source term were modeled following Candlerand MacCormack.11 For C02, the following reactionswere modeled:
CO2 + M v* CO + 0 + M
CO + 0CO2 + 0
02 + C02 + CO
where the equilibrium constants, transport properties,and translational-vibrational energy exchange sourceterm were modeled following Rock et al.12 and Parket a/.13 An approximation was made, however, in mod-eling the CO2 vibrational energy. Carbon dioxide hasthree vibrational modes, one of which is doubly de-generate. This can be approximated by a four-fold de-generate vibrational mode characterized by the lowestvibrational energy constant. While not giving exactlythe same quantitative results, this approximation stillcaptures the qualitative behavior of the vibrational en-ergy term.
B. Numerical Method
The Navier-Stokes equations for axisymmetric flowwere solved with a -finite- volume method using modi-fied Steger- Warming flux-vector splitting.14 Althoughfirst-order accurate normal to the wall, MacCormackand Candler14 have shown that the modified Steger-Warming flux-vector splitting method is less dissipa-tive as compared to true Steger- Warming in regions ofhigh shear and yields accurate profiles in hypersonicflows compared to boundary layer methods.
The mean flow equations were solved on a massivelyparallel Thinking Machines CM-5 using the implicitData-Parallel Lower-Upper Relaxation method.15 Thescheme is first order accurate in the body-normaldirection and second order in the streamwise direc-tion. Freestream conditions were chosen for compar-ison with experimental results obtained by Germainand Hornung7 and Adam and Hornung.8 Freestreamconditions for three cases for which detailed results arepresented are given in Table 1, where th§ densities areordered as:
Ps = (pco2 , Pco ,Po2,pc,Po) , for CO2 , andPs =(PN2,P02,PNO,PN,PO) ,for air.
For each case, the body was a cone with a half-angleof 5° at zero incidence. The length of the T5 conewas 0.993 m while the length of the computationalmodel was 3 m to insure that it was long enough thattransition would be predicted on the body.
Table 1. Freestream parameters for Shots 1150,1157,and 1162.
C. Mean Flow Boundary Conditions
Boundary conditions were prescribed at each of thesolution domain boundaries: the wall, the freestream,the inflow, and the outflow. In each case, the wall wasconsidered no-slip and assumed isothermal at a tem-perature of 293 K so that vibrational and translationaltemperatures at the wall were equal to the wall tem-perature. The wall was assumed non-catalytic suchthat there were zero mass concentration gradients nor-mal to the surface, and a zero pressure gradient normalto the wall was also assumed. The inflow and outfloware supersonic, except in the boundary layer. Thus,the inflow conditions were specified from the experi-mental conditions and a zero-gradient boundary con-dition was applied at the outflow boundary. Zero gra-dient was assumed along the top boundary in all theflow variables to allow relaxation of the nonequilibriumfreestream conditions at downstream locations.
D. Mean Flow Grid
Since stability analysis requires an accurate meanflow solution, the mean flow grid must be fine enoughto capture the key physical aspects of the flow. Inthis work, grid points were clustered near the tip ofthe cone by exponential stretching in the streamwisedirection, thus ensuring an accurate solution of theearly growth of the boundary layer. In the surface-normal direction, the grid was stretched exponentiallyaway from the wall with a minimum surface normalspacing chosen such that the wall variable, y+, wasless than one in the first cell over most of the body.This typically required using a minimum spacing of1 x 10~6 m, and resulted in approximately 100 pointsclustered within the boundary layer. In addition, thegrid was generated normal to the body surface so thatboundary layer profiles at various streamwise locationscould be used by the stability code without interpola-tion. In each case, the mean flow was computed using
256 points in both the streamwise and body-normaldirections.
A grid convergence study was performed in whicha 256 x 512 grid was used to compute the mean flow.Since the results of a stability analysis of the flow com-puted on this more refined grid were very close to thosecomputed on the 256 x 256 grid, it was decided that the256 x 256 grid was the most computationally efficientgrid size to use that would not result in significantloss in accuracy of the solution. Using this grid, cal-culations of laminar heat transfer rates matched theexperimentally measured values very well up to thepoint of transition.
III. LINEAR STABILITY
The equations describing the stability of a gas flowin thermal and chemical nonequilibrium are derivedfrom the Navier-Stokes equations. The instantaneousflow is modeled by a mean and fluctuating component,q = q + q', where q is any flow variable such as speciesdensity, velocity, or temperature. This perturbationis substituted into the Navier-Stokes equations whichare then linearized with respect to the fluctuations andthe mean flow is subtracted. Two assumptions areemployed in the derivation of the linear stability equa-tions: linearity and parallel flow.
Linearity implies that there is no interaction be-tween the mean flow and the disturbances. The fluc-tuating disturbance is assumed to be small comparedto the mean flow, such that higher order fluctuatingterms are neglected and the resulting equations are lin-ear. The linear assumption has been questioned due tothe nonlinear processes leading to transition and theessentially nonlinear nature of turbulence. However, ithas been noted16 that the factors that affect the linearamplification are the primary factors that determinethe magnitude of the transition Reynolds number be-cause the linear amplification step is the slowest of themultiple steps of transition. Also, the frequency of themost amplified disturbance in the linear range is theone that persists into the nonlinear range.6
The parallel flow assumption states that the meanflow is assumed to vary only in the body normal di-rection. This assumption, which is used only in thestability equations and not in the computation of themean flow, is valid for small boundary layer growthover a wavelength of the instability wave. Changet al.17 have concluded that flow nonparallelism hasmore influence on oblique first-mode disturbances thanon two-dimensional second-mode disturbances. Whilethe parabolized stability equations do not employ thelinear or parallel flow assumptions, they are computa-tionally more demanding to solve. The linear stabil-
ity theory has been generally accepted to yield quali-tatively correct parametric stability relations. Muchwork has been done by Hudson to develop a com-putational tool for analyzing boundary layer stabil-ity in thermal and chemical nonequilibrium.18 Themethods and results have been extensively tested andverified through comparison with the results of otherresearchers.3'18 That work provided the basis for muchof the current work.
A. Stability Equations
The stability equations are obtained by assumingthat the perturbations or fluctuations are describedby a normal mode:
The mean flow is assumed parallel such that q(x, y, z) =q(y) and the mean flow velocity component in thewall-normal direction is assumed equal to zero. Thespecies diffusion velocity in the streamwise direction isassumed to be zero, but the species diffusion velocityin the normal direction is not restricted. Variation ofquantities in the azimuthal direction are also assumedto be zero.
For spatial stability, the frequency is real and thewave numbers are complex such that a = ar + ictiand we assume the azimuthal wave number,/?, is real.The disturbances amplify in space and are unsta-ble if — cti > 0. The waves travel with phase ve-locity, CT = u/ar, at an orientation angle given byi/) = ta.n~l(/3/ar). For two-dimensional waves, ft iszero.
The stability equations are obtained by substitut-ing the normal mode fluctuation into the linearizedperturbation equations. The result is a system of tenordinary differential equations which can be written inmatrix form as:
where (j> = (ps,u,v,w,T,Tv), ps,s = 1...5 are thespecies mass densities, and u, v, w are the disturbancevelocities in the x, y, and z directions, respectively. A,B, and C are 10 x 10 complex matrices which are func-tions of the mean flow, a, 0, and w and are presentedin detail in Hudson.18
B. Linear Stability Boundary ConditionsThe stability analysis requires boundary condi-
tions on disturbances both at the wall and in thefreestream. The no-slip condition at the wall meansthat the velocity fluctuations at the wall are zero.
Figure 1. Convergence of amplification rates withincreasing number of local stability grid points.
The temperature fluctuations at the wall are assumedto be zero due to the damping effect of the thermalmass of the body. The fluctuations of species densitysatisfy the non-catalytic wall condition.
A numerical boundary condition is required at thefreestream boundary to avoid a trivial solution. Fol-lowing the work of Stuckert,19 all disturbances exceptthe disturbance velocity in the y-direction, v, are as-sumed zero in the freestream. The mixture mass con-servation equation is applied to obtain an expressionfor the value of v at the freestream boundary.
C. Stability Grid
The grid which was chosen for the stability compu-tations is cosine-exponential. First, cosine stretchingwas used to place half of the total number of pointswithin the boundary layer with clustering both at thewall and at the boundary layer edge. Exponentialstretching was then used to place the second half of thepoints between the boundary layer edge and the outercomputational boundary which was located sufficientlyfar from the edge that freestream conditions could beapplied. For supersonic or hypersonic flow, a shockforms between the boundary layer and the freestream.It has been shown that the shock has little effect onthe boundary layer stability when the location of theshock is far outside the boundary layer5. When theoutermost points of the stability grid crossed a shock,the flow conditions from just inside the shock were ap-plied to all the grid points from the shock locationto the edge of the computational domain. This wasdone to prevent possible numerical oscillations causedby the presence of the shock, but did not affect theaccuracy of the solution.
Grid convergence studies were performed to ensurethat the computed eigenvalues were independent of thestability grid. As an example, Fig. 1 shows amplifica-tion curves computed with the number of stability gridpoints ranging from 125 to 400. Here, the lowest num-ber of grid points which ensures an accurate solutionis 300. For all of the results presented in this paper,300 points were used in the local stability grid for nor-mal cases, and 400 points were used when the heats offormation were reversed.
The stability equations (4) constitute an eigenvalueproblem for which non-zero solutions exist only for dis-crete values of wavenumber and frequency. Followingthe work of Malik,20 the eigenvalue problem was solvedin two steps. The first step is a global method whichyields a spectrum of approximate eigenvalues. The ap-proximate eigenvalues are then used as an initial guessfor the local method which uses iterative refinementto converge to the most unstable eigenvalue. To speedup the solution, the global method was not run at ev-ery point. Instead, previous solutions were rescaled inboth frequency and wavenumber and used as guessesin the local method. The linear stability code was runon multiprocessor SGI workstations.
IV. Boundary Layer Transition Prediction
Linear stability theory has been used for many yearsto analyze boundary layer stability, but only recentlyhas it been used to predict transition. The eN methodexamines the amplitude growth of constant frequencydisturbances, and is defined as:
or equivalently,
X
/I dA— -—dxA dx
Xo
X
(5)
where x0 is the first neutral point at a given frequency,Jt^z' ls *^e sPati&l amplification rate, — a^, and theintegration is performed at a constant physical fre-quency, ui. This integration is carried out until theend of the body is reached or the disturbance at thefrequency u> becomes stable. The value of AT at transi-tion has been experimentally shown to be about 10 formany different geometries and freestream conditions.21
The frequency which produces N = 10 at the first x-location is the most unstable frequency, and transitionis deemed to occur at that location.
Germain and Hornung7 suggested that for transi-tion analysis, the Reynolds number should be com-puted with the material properties evaluated at the
reference temperature as the behavior of laminarboundary layers may then be rendered almost inde-pendent of Mach number, specific heat ratio, and walltemperature ratio. That is,
Re'tr =P UeXtr
(6)
where p* and /z* are the density and viscosity eval-uated at the reference temperature, T*, assumingp = p*, ue is the local edge velocity, and xtr is thesurface transition location. The reference temperaturefor air is given by the formula:22
(7)
where Me is the edge Mach number, Te is the edgetemperature, and Tw is the wall temperature.
V. Results and Discussion
The validated mean flow and stability codes wereused to evaluate the effect of freestream enthalpy onthe transition Reynolds number. Each case which wastested for air, nitrogen, and carbon dioxide used a dif-ferent set of freestream conditions which were chosento match those of the experiments. The mean flowwas computed and analyzed with the stability code,and the eN method was used to predict the transitionlocation.A. Boundary Layer Thicknessand Most Amplified Frequency
Mack23 showed that when M > 1, the disturbanceequation becomes approximately a wave equation inthe region between the relative sonic line, M =1,and the body surface. Lees and Gold24 suggested thatthe second and higher mode disturbances are the re-sult of sound waves which reflect back and forth inthis region, and Morkovin25 showed, using a geomet-rical argument, which disturbance frequencies wouldbe amplified. Using this argument, we can define acharacteristic time, T, as the time it takes for a distur-bance to travel from the wall to the relative sonic line,located at a distance ya from the surface, as:
J/o
0
(8)
1.5
1.25
0.75
0.5
D AirO N2
111
10 15h0 [MJ/kg]
Figure 2. Product of most amplified disturbance fre-quency and characteristic time.
Then a disturbance should be amplified when its fre-quency is given by:
TLJ—— = n , for n = 1,3,5, •7T
(9)
where n = 1 corresponds to a second-mode distur-bance.
Figure 2 shows values of TU/TT at the most un-stable frequency for each flow that was studied ateach location along the body. As predicted by thetheory, for second mode disturbances in flows of lowtotal enthalpy, the parameter TU/TT is very close toone. As the freestream total enthalpy increases, thisparameter first increases then decreases for both airand nitrogen. The larger change with enthalpy in thecase of air is presumably related to the lower dissoci-ation energy of oxygen present in the gas mixture andshows the increasing importance of chemical reactionsat higher enthalpies. At ho < 6 MJ/kg, all of themost unstable frequencies have a value of the param-eter TW/TT > 1.0. As we move through the enthalpyrange of /i0 = 6 — 10 MJ/kg, the parameter TLJ/TT atthe most unstable frequencies shifts to lower values.Further work will need to be done to fully understandwhat is causing this shift, and careful numerical ex-periments may lead to the answers.
While the S-LJ relationship discussed above mayhelp us understand the unstable disturbance frequen-cies, it assumes that we already have the solution andtherefore know the location ya. What is desired isan engineering method of predicting the most unsta-ble frequencies. Various researchers have shown rela-tionships between disturbance frequency and bound-ary layer thickness for second-mode disturbances.
FigureS. Dimensional product of most amplified dis- Figure 5. Amplification rates for reacting distur-turbance frequency and boundary layer thickness. bances in reacting flow, Shot 1157.
N
~3H*
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
-
-
| „
§ J l| |i|l| 1A J fc IT i
| Br •
-
- a Ah-; ON,-^ , , , , i , , , , i , . , ,
15
h0 [MJ/kg]
Figure 4. Product of most amplified disturbance fre-quency and boundary layer thickness, nondimension-alized by reference speed of sound.
For the air and nitrogen flows, Fig. 3 shows the re-lationship between the product of the most unstablefrequency at a given surface location and the bound-ary layer thickness, 8u, and the freestream enthalpywhere 6cj has units of kHz • m. This increase in Su>with freestream enthalpy was first noted by Seipp.26
An approximate method of removing the enthalpydependence is to nondimensionalize uj by multiplyingit by a characteristic time based on the boundary layeredge thickness,
Te = 2 / ad2/ (10)
3000
2500
2000
1500
1000
500
Stable
Transition at x= 0.6 m
Stable
0.5 1.5x[m]
2.5
Figure 6. Amplification rates for reacting distur-bances in reacting flow, Shot 1157.
to travel from the wall to the boundary layer edge, 6.This works well if one has a boundary layer solutionprovided by a CFD analysis, and it was used in thiswork to estimate u and rescale guesses in the localanalysis. In other cases, an approximate characteristictime can be found using the boundary layer thicknessand the reference temperature speed of sound, a*, as
T' = 2o*
where re is half the time required for a sound wave
The results shown in Fig. 4 demonstrate that thismethod approximately removes the enthalpy depen-dence. However, there is still the interesting behaviorin the air data at ft0 = 6 — 10 MJ/kg.
Figure 7. Variation of transition Reynolds numberand transition frequency with value of N selected astransition value, Shot 1157.
B. Effect of Freestream Enthalpyon Transition Location
Figure 5 shows disturbance amplification rates forShot 1157. This is a typical case at a total enthalpy ofho — 6.12 MJ/kg. The disturbance amplification ratessmoothly increase along the body to a maximum valueof approximately — a» = 85.8/m at a surface locationof x = 0.088 m and then begin to decrease. This sameinformation can also be presented with a contour plotof amplification rate versus disturbance frequency andsurface distance. This is shown in Fig. 6. Using the e^method with N — 10 and the information in this plot,transition is predicted at x = 0.599 m, correspondingto Retr = 2.04 x 106 (Retr = 4.20 x 106). From thesegraphs we can see that while the maximum amplifica-tion rate occurs at a frequency of 2576 kHz, such highfrequency disturbances are only unstable over a verysmall portion of the body, and the integrated ampli-fication rate does not have a chance to become verylarge. Rather, transition is predicted for disturbancesat the lower frequency of 1162 kHz. A similar anal-ysis was performed for each of the air, nitrogen, andcarbon dioxide flows which were tested.
Using Shot 1157 as an example, Fig. 7 shows howthe transition Reynolds number and most unstablefrequency vary with the transition value chosen forN. With N = 10, transition is predicted at Re$r =2.04 x 106 while the experimentally measured tran-sition Reynolds number was Re%r = 0.735 x 106,occurring at N = 4.39. Table 2 shows the ra-tios of the computed to the experimental transitionReynolds numbers and the values of N which would
GasAirAirAirAirAirAirAirN2N2N2N2N2N2
ho [MJ/kg]3.356.128.0710.611.712.713.53.256.007.709.6811.212.3
Table 2. Comparison between computed and experi-mental transition locations.
AirN,
10 15
h0[MJ/kg]
Figure 8. Transition edge Reynolds number, Retr,vs. freestream total enthalpy.
give the experimentally measured values of Retr. Ex-periments in quiet wind tunnels and flight tests21'27
have shown that the eN method gives reliable tran-sition predictions when the value of N is approxi-mately 9 — 11. The values of N which would havegiven transition locations matching the experimentalresults are w 4 - 7 which is small compared to whathas been computed for transition in "quiet" facilitiesor in flight experiments.21'27 Germain and Hornung7
mention that high freestream noise levels may pro-vide a non-linear bypass mechanism for transition inthe T5 shock tunnel. This could help to explain whylinear stability consistently overpredicts the transitionReynolds numbers.
Figure 8 shows a plot of transition edge Reynoldsnumber, Retn versus freestream total enthalpy for
Figure 9. Transition Reynolds number, Relr, vs.freestream total enthalpy for air. Experimental datafrom[7] and[8].
so
40
30
20
10
0
-10
-20
-30
-40
-50,
x = 0.943 m x = 0.316 m
1000 2000
co [kHz]3000 4000
Figure 11. Amplification rates for frozen distur-bances in reacting air flow, Shot 1162.
10'r
£. 10°
0)DC
10'1
o
• N2, computationalO N,, experimental
5 10
h0[MJ/kg]15
Figure 10. Transition Reynolds number, Refr, vs.freestream total enthalpy for nitrogen. Experimentaldata from[7] and[8).
each case. As was noted by other researchers,8'28
there are no discernible trends when the data is plot-ted in this manner. Figure 9 shows a plot of tran-sition Reynolds number versus freestream total en-thalpy for air flows, where now the Reynolds numberis calculated at the reference temperature conditions.Both the computational results and those from theexperiments7'8 are shown. While the predicted transi-tion locations are about a factor of 1.9 — 2.8 largerthan what was measured, the numerical simulationproduces strikingly similar results. That is, over therange of enthalpies which were tested, Re^r seems toincrease exponentially with /i0.
50
40
30
20
-, 10
C. 0
-20
-30
-40
-50
x= 0.943 m x= 0.316mIncreasing x
1000 2000
co [kHz]3000
Figure 12. Amplification rates for reacting distur-bances in reacting flow, Shot 1162.
Figure 10 shows the same plot for nitrogen. Whilethe computed transition locations are about a factor of1.4 — 2.0 above those which were measured experimen-tally, the numerical results show the same trend thatthe increase in Re%r with ho is much smaller comparedto air flows. A slight increase in Refr is apparent inthe graph, but because of the larger dissociation en-ergy of N2, we would not expect to see a significantincrease in Refr until higher freestream enthalpies.
In all the CC>2 cases which were tested, the bound-ary layers were either stable, or transition was not pre-dicted by end of the body. Clearly, there is a strongdestabilizing effect of the different gas models on the
Figure 13. Amplification rates for reacting distur-bances in reacting flow. Heats of formation have beenreversed in the disturbances, Shot 1162.
boundary layer transition location. We will demon-strate that this effect is related to the endothermicityof reactions which occur in the different gases as thefreestream enthalpy is increased.
C. Effect of Chemical Reactions in Disturbances
Running numerical experiments gives us the uniqueopportunity to manipulate the gas parameters in or-der to explore how they affect the stability. To iso-late the effects of thermochemical nonequilibrium inthe disturbances, we can start with the nonequilib-rium mean flow solution and freeze the disturbancessuch that they maintain the mean chemical composi-tion and distribution of internal energy. This is doneby multiplying both the chemical reaction rates andthe translational-vibrational relaxation rate by a fac-tor of 10~4 in the stability code while the mean flow so-lutions are left unchanged. Running the stability codefor Shot 1162 with the disturbances frozen in their lo-cal chemical state, we get the results shown in Fig. 11.Comparing this with the results shown in Fig. 12 forreacting disturbances, we can see that at different lo-cations in the same mean flow, the effect of chemistryand translational-vibrational energy transfer can be ei-ther stabilizing or destabilizing. Since the mean flowwas unchanged between these two examples, any dif-ferences in the disturbance amplification rates are dueto the effect of chemistry in the disturbances alone.As a measure of overall boundary layer stability, wecan compute the transition Reynolds number for thisflow. This gives Re*tr = 3.53 x 106 (Retr = 4.08 x 106)when the disturbances are allowed to react, but onlyRe;r = 2.16 x 106 (Retr = 3.36 x 106) when the distur-
bances are frozen. This demonstrates the stabilizingeffect of the chemistry.
We can perform another interesting experiment ifinstead of freezing the chemical composition, we switchthe signs of the heats of formation in the disturbances.In this manner, a dissociation reaction which was pre-viously endothermic is now exothermic. Figure 13shows, for Shot 1162, amplification rates for react-ing disturbances in reacting flow, where the heats offormation are reversed in the disturbances. In gen-eral, these disturbances have much larger amplifica-tion rates compared to either the frozen or the en-dothermic cases. Compared to the endothermic casewhere reactions stabilized the disturbances, the flowis now more unstable, and transition is predicted atRe*tr = 1.68 x 106 (Retr = 2.71 x 106). This showsthat there is a strong effect of heat release on the dis-turbance amplification rates.
In the examples shown above, the mean flow itself ischemically reacting and the reactions may, at differentsurface locations, be moving in the direction of eitherdissociation or recombination of molecules. Comparedwith a perfect gas flow, the different stability charac-teristics are the result of a combination of changes bothin the mean flow profiles and in the disturbances. Tostudy this, the freestream conditions of Shot 1162 wereused to compute a perfect gas mean flow. Then, run-ning the stability code with frozen disturbances in thisfrozen mean flow we get the results shown in Fig. 14.In this case, transition is predicted at Re^r = 2.44 x 106
(Retr = 4.52 x 106) and the boundary layer is morestable than that for the reacting mean flow case above(Fig. 11) where for frozen disturbances in the reactingmean flow, transition is predicted at Re%r = 2.16 x 106
(Retr - 3.36 x 106). Thus, in this case, the effectof chemistry in establishment of the mean flow is todestabilize the boundary layer relative to the perfectgas case.
Next, we run the stability code on the perfect gasmean flow, but allow the disturbances to chemicallyreact. These results are shown in Fig. 15. Allow-ing chemical reactions in the disturbances dramati-cally stabilizes the boundary layer. As seen from Ta-ble 1, the perfect gas mean flow consists primarily ofdiatomic species. Once the frozen mean flow is es-tablished, any reactions that are allowed can, for themost part, only proceed in the direction of dissocia-tion and are therefore endothermic. Thus, the pres-ence of endothermic reactions stabilizes the fluctua-tions, and transition is not predicted by the end of thebody. A summary of transition Reynolds numbers forthe different variations of Shot 1162 which were tested
Figure 14. Amplification rates for frozen distur- Figure 16. Amplification rates for frozen distur-bances in frozen air flow, Shot 1162. bances in reacting COa, Shot 1150.
20
0
-20
-40
¥n. -sotr
-80
-100
-120
-140
X= 0.943 mx= 0.316m
viy Increasing x
x=0.106 m
Unstable
A
\lStable
A__0 1000 2000
co [kHz]3000 4000
Figure 15. Amplification rates for reacting distur-bances in frozen air flow, Shot 1162.
Mean flowReactingReactingReactingPerfect gasPerfect gas
DisturbancesReactingFrozen
Opposite hoFrozen
Reacting
fle(*r/106
3.532.161.682.44None
-Retr/106
4.083.362.714.52None
Table 3. Transition Reynolds numbers for variationson Shot 1162.
are given in Table 3.Similar behavior is observed for the carbon dioxide
flow of Shot 1150. The chemically reacting mean flowwas computed and a stability analysis was performedwith the disturbances frozen in their mean
-100
-200
-300
-400
-500
x= 0.936m x= 0.316m
Stable
Increasing x
1000 2000
a [kHz]3000 4000
Figure 17. Amplification rates for reacting distur-bances in reacting COa, Shot 1150.
chemical state. The results are shown in Fig 16. Next,the disturbances were allowed to experience chemicalreactions and translational-vibrational energy trans-fer, and the nonequilibrium disturbance amplificationrates were computed. These results are shown inFig. 17. While the frozen disturbances were quite un-stable, and transition is predicted at Re^T = 4.44 x 106
(Retr = 4.37 x 106), allowing the disturbances to reactcompletely stabilizes the boundary layer.
VII. Conclusions
In this paper, linear stability and the eN methodare shown to predict transition locations which are ingeneral agreement with experimental data. The com-puted transition Reynolds numbers are about a factor
of two larger than what was experimentally measured -a little more for air and a little less for nitrogen. Moreimportantly, we reproduce the same trends that wereobserved in the experiments.
It has been hypothesized that the enthalpy ef-fects on transition are due to the increasing impor-tance of chemical reactions in the boundary layers athigher freestream enthalpies.7 The increase in transi-tion Reynolds number for air flows is more prominentthan it is for nitrogen, with the hypothesized reasonthat the lower dissociation energy of 62 causes it tomore easily absorb fluctuations in the boundary layer,lowering the disturbance amplification rates. Carbondioxide flows, which have still lower dissociation ener-gies and can more easily absorb energy fluctuations,are much more stable than either those of air or ni-trogen. These results were also observed in the linearstability analysis.
The combination of CFD and linear stability anal-ysis provides a unique tool by which various aspectsof the chemistry effects can be isolated and subjectedto study in a way that is not possible in physical ex-periments. As freestream enthalpy is increased, boththe mean flow profiles and the disturbances change aschemical reactions become important. In computa-tional models, we can separate these effects to studytheir relative importance. We can perform unique ex-periments by computing perfect gas or reacting meanflows, by freezing the local chemical composition ofdisturbances, or by switching the heats of formationto study the effects on the stability. In doing so, itwas demonstrated that the presence of chemical reac-tions can play a large role in disturbance amplificationrates. First, it was found that the effect of chemi-cal reactions in the establishment of the mean flowprofiles leads to a more unstable boundary layer com-pared with the perfect gas case. However, it was alsofound that the presence of endothermic reactions inthe disturbances can be stabilizing, while exothermicreactions are destabilizing. Thus, there are competinginfluences of chemistry in different aspects of the flowstability.
Acknowledgments
The authors would like to thank Philippe Adamat GALCIT for providing freestream conditions andtransition data for the T5 experiments.
Support for the authors is provided by the ArmyResearch Office under Grant No. DAAH04-95-1-0540and AASERT Grant No. DA/DAAH04-96-1-0269.This work was also sponsored in part by the ArmyHigh Performance Computing Research Center under
the auspices of the Department of the Army, ArmyResearch Laboratory cooperative agreement numberDAAH04-95-2-0003 / contract number DAAH04-95-C-0008, the content of which does not necessarily re-flect the position or the policy of the government, andno official endorsement should be inferred.
References
[1] M. R. Malik and E. C. Anderson. "Real Gas Ef-fects on Hypersonic Boundary-Layer Stability".Physics of Fluids A, 3(5):803-821, May 1991. •
[2] G. K. Stuckert and H. L. Reed. "Linear Distur-bances in Hypersonic, Chemically Reacting ShockLayers". AIAA Journal, 32(7), July 1994.
[3] M. L. Hudson, N. Chokani, and G. V. Candler."Linear Stability of Hypersonic Flow in Thermo-chemical Nonequilibrium". AIAA Journal, 35(6),June 1997.
[4] M. L. Hudson, N. Chokani, and G. V. Candler."Nonequilibrium Effects on Hypersonic BoundaryLayers and Inviscid Instability". AIAA Paper No.94-0825, January 1994.
[5] C. L. Chang, M. R. Malik, and M. Y. Hussaini."Effects of Shock on the Stability of HypersonicBoundary Layers". AIAA Paper No. 90-1448,June 1990.
[6] Leslie M. Mack. "Linear Stability Theory and theProblem of Supersonic Boundary-Layer Transi-tion". AIAA Journal, 13(3):278-289, March 1975.
[7] P. D. Germain and H. G. Hornung. "Transitionon a Slender Cone in Hypervelocity Flow". Ex-periments in Fluids, 22(3), January 1997.
[8] Philippe H. Adam and Hans G. Hornung. "En-thalpy Effects on Hypervelocity Boundary LayerTransition: Experiments and Free Flight Data".AIAA Paper No. 97-0764, January 1997.
[9] M. P. Martin and G. V. Candler. "Effect of Chem-ical Reactions on the Decay of Isotropic Homoge-neous Turbulence". AIAA Paper No. 96-2060,June 1996.
[10] L. Lees and C. C. Lin. Investigation of the Sta-bility of the Laminar Boundary Layer in a Com-pressible Flow. NACA Technical Note 1115,1946.
[11] G. V. Candler and R. W. MacCormack. "TheComputation of Hypersonic Ionized Flows inChemical and Thermal Nonequilibrium". Journalof Thermophysics and Heat Transfer, 5(3):266-273, July 1991.
[12] S. G. Rock, G. V. Candler, and H. G. Hor-nung. "Analysis of Thermocheraical Nonequilib-rium Models for Carbon Dioxide Flows". AIAAJournal, 31(12):2255-2262, December 1993.
[13] C. Park, J. T. Howe, R. L. Jaffe, and G. V. Can-dler. "Review of Chemical-Kinetic Problems ofFuture NASA Missions, II: Mars Entries". Jour-nal of Thermophysics and Heat Transfer, 8(1),January-March 1994.
[14] R. W. MacCormack and G. V. Candler. "TheSolution of the Navier-Stokes Equations UsingGauss-Seidel Line Relaxation". Computers andFluids, 17(1):135-150, 1989.
[15] G. V. Candler, M. J. Wright, and J. D. McDonald."Data-Parallel Lower-Upper Relaxation Methodfor Reacting Flows". AIAA Journal, 32(12):2380-2386, December 1994.
[16] E. Reshotko. "Remark on Engineering Aspects ofTransition". In Richard E. Meyer, editor, Tran-sition and Turbulence, page 147. Academic Press,1981.
[17] C. L. Chang, M. R. Malik, G. Erlebacher, andM. Y. Hussaini. "Compressible Stability of Grow-ing Boundary Layers Using Parabolized StabilityEquations". AIAA Paper No. 91-1636, June 1991.
[18] M. L. Hudson. Linear Stability of HypersonicFlows in Thermal and Chemical Nonequilibrium.PhD thesis, North Carolina State University,February 1996.
[19] G. K. Stuckert. Linear Stability Theory of Hyp-ersonic, Chemically Reacting Viscous Flow. PhDthesis, Arizona State University, December 1991.
[20] M. R. Malik. "Numerical Methods for HypersonicBoundary Layer Stability". Journal of Computa-tional Physics", 86:376-413, 1990.
[21] N. A. Jaffe, T. T. Okamura, and A. M. O. Smith."Determination of Spatial Amplification Factorsand Their Application to Predicting Transition" .
Journal, 8(2), February 1970.
[22] E. R. G. Eckert. "Engineering Relations for Fric-tion and Heat Transfer to Surfaces in High Veloc-ity Flow". Journal of the Aeronautical Sciences,22:585-587, 1955.
[23] L. M. Mack. "Boundary-Layer Linear StabilityTheory," Special Course on Stability and Transi-tion of Laminar Flow. AGARD Report 709, June1984.
[24] L. Lees and H. Gold. "Stability of Lami-nar Boundary Layers and Wakes at HypersonicSpeeds. I - Stability of Laminar Wakes". In In-ternational Symposium on Fundamental Phenom-ena in Hypersonic Flow, Pasadena, CA, January1966. California Institute of Technology.
[25] Mark V. Markovin. "Transition at HypersonicSpeeds". NASA Contractor Report 178315, May1987.
[26] T. G. Seipp. "The Effect of Freestream Enthalpyon Hypersonic Boundary Layer Stability". Mas-ter's thesis, University of Minnesota, December1996.
[27] M. R. Malik. "Prediction and Control of Tran-sition in Hypersonic Boundary Layers". AIAAPaper No. 87-1414, June 1987.
[28] P. Germain. The Boundary Layer on a SharpCone in High-Enthalpy Flow. PhD thesis, Grad-uate Aeronautical Laboratories, California Insti-tute of Technology, 1994.
![Page 1: [American Institute of Aeronautics and Astronautics 32nd Thermophysics Conference - Atlanta,GA,U.S.A. (23 June 1997 - 25 June 1997)] 32nd Thermophysics Conference - Numerical study](https://reader035.documents.pub/reader035/viewer/2022081215/575094a51a28abbf6bbadc66/html5/thumbnails/1.jpg)
![Page 2: [American Institute of Aeronautics and Astronautics 32nd Thermophysics Conference - Atlanta,GA,U.S.A. (23 June 1997 - 25 June 1997)] 32nd Thermophysics Conference - Numerical study](https://reader035.documents.pub/reader035/viewer/2022081215/575094a51a28abbf6bbadc66/html5/thumbnails/2.jpg)
![Page 3: [American Institute of Aeronautics and Astronautics 32nd Thermophysics Conference - Atlanta,GA,U.S.A. (23 June 1997 - 25 June 1997)] 32nd Thermophysics Conference - Numerical study](https://reader035.documents.pub/reader035/viewer/2022081215/575094a51a28abbf6bbadc66/html5/thumbnails/3.jpg)
![Page 4: [American Institute of Aeronautics and Astronautics 32nd Thermophysics Conference - Atlanta,GA,U.S.A. (23 June 1997 - 25 June 1997)] 32nd Thermophysics Conference - Numerical study](https://reader035.documents.pub/reader035/viewer/2022081215/575094a51a28abbf6bbadc66/html5/thumbnails/4.jpg)
![Page 5: [American Institute of Aeronautics and Astronautics 32nd Thermophysics Conference - Atlanta,GA,U.S.A. (23 June 1997 - 25 June 1997)] 32nd Thermophysics Conference - Numerical study](https://reader035.documents.pub/reader035/viewer/2022081215/575094a51a28abbf6bbadc66/html5/thumbnails/5.jpg)
![Page 6: [American Institute of Aeronautics and Astronautics 32nd Thermophysics Conference - Atlanta,GA,U.S.A. (23 June 1997 - 25 June 1997)] 32nd Thermophysics Conference - Numerical study](https://reader035.documents.pub/reader035/viewer/2022081215/575094a51a28abbf6bbadc66/html5/thumbnails/6.jpg)
![Page 7: [American Institute of Aeronautics and Astronautics 32nd Thermophysics Conference - Atlanta,GA,U.S.A. (23 June 1997 - 25 June 1997)] 32nd Thermophysics Conference - Numerical study](https://reader035.documents.pub/reader035/viewer/2022081215/575094a51a28abbf6bbadc66/html5/thumbnails/7.jpg)
![Page 8: [American Institute of Aeronautics and Astronautics 32nd Thermophysics Conference - Atlanta,GA,U.S.A. (23 June 1997 - 25 June 1997)] 32nd Thermophysics Conference - Numerical study](https://reader035.documents.pub/reader035/viewer/2022081215/575094a51a28abbf6bbadc66/html5/thumbnails/8.jpg)
![Page 9: [American Institute of Aeronautics and Astronautics 32nd Thermophysics Conference - Atlanta,GA,U.S.A. (23 June 1997 - 25 June 1997)] 32nd Thermophysics Conference - Numerical study](https://reader035.documents.pub/reader035/viewer/2022081215/575094a51a28abbf6bbadc66/html5/thumbnails/9.jpg)
![Page 10: [American Institute of Aeronautics and Astronautics 32nd Thermophysics Conference - Atlanta,GA,U.S.A. (23 June 1997 - 25 June 1997)] 32nd Thermophysics Conference - Numerical study](https://reader035.documents.pub/reader035/viewer/2022081215/575094a51a28abbf6bbadc66/html5/thumbnails/10.jpg)
![Page 11: [American Institute of Aeronautics and Astronautics 32nd Thermophysics Conference - Atlanta,GA,U.S.A. (23 June 1997 - 25 June 1997)] 32nd Thermophysics Conference - Numerical study](https://reader035.documents.pub/reader035/viewer/2022081215/575094a51a28abbf6bbadc66/html5/thumbnails/11.jpg)
![Page 12: [American Institute of Aeronautics and Astronautics 32nd Thermophysics Conference - Atlanta,GA,U.S.A. (23 June 1997 - 25 June 1997)] 32nd Thermophysics Conference - Numerical study](https://reader035.documents.pub/reader035/viewer/2022081215/575094a51a28abbf6bbadc66/html5/thumbnails/12.jpg)
![Page 13: [American Institute of Aeronautics and Astronautics 32nd Thermophysics Conference - Atlanta,GA,U.S.A. (23 June 1997 - 25 June 1997)] 32nd Thermophysics Conference - Numerical study](https://reader035.documents.pub/reader035/viewer/2022081215/575094a51a28abbf6bbadc66/html5/thumbnails/13.jpg)
