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FOREWORD
This final report presents work conducted for the Marshall Space Flight Center(MSFC), National Aeronautics and Space Administration, in response to the requirementsof Contract NAS8-38185. The work presented here was performed by REMTECH inc.,Huntsville, AL, and is titled "External Tank Aerothermal Design Criteria Verification."
The project manager for this project was Dr. Sarat C. Praharaj. The NASA contractmonitor for this project was Mr. Lee D. Foster, ED33, of the Induced Environments
Branch of the Aerophysics Division.
F:__'rvl-r" _" _ _--I RTR 227-01
ABSTRACT
If an SSME fails during the initial 160 seconds of the Shuttle flight, a return-to-launch-site maneuver will be implemented. The period of concern for this task is the pitch-aroundmaneuver when the vehicle is flying backward. The intent of this report is to identify anddefine the flowfield at the most critical locations from an environment perspective. Thesolution procedure used to predict the plume heating rates involves both computationalanalysis and engineering modeling.
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Contents
FOREWORD ............................................. i
ABSTRACT ............................................. ii
List of Figures ........................................... ivList of Tables ............................................ iv
Nomenclature ............................................ v
1 INTRODUCTION ......................................... 1
2 MODELING FOR ABORT CASES .............................. 6
2.1 Trajectories .......................................... 6
2.2 CFD Modeling ........................................ 6
2.3 Engineering Modeling (Heating Indicator Approach) ............... 133 GENERATION OF ABORT ENVIRONMENTS ..................... 16
4 CONCLUSIONS AND RECOMMENDATIONS ..................... 21
4.1 Conclusions ........................................ 21
4.2 Recommendations .................................... 21
5 REFERENCES ......................................... 23
Appendix A Two-Species Variable Gamma Formulation for a Forward-FiringRocket Nozzle ................................... 24
Appendix B Variable Gamma Computations for a Forward-Firing Space ShuttleMain Engine .................................... 52
iii
F_"M'-FE_:C V-I RTR 227-01
List of Figures
1 Return-to-Launch-Site (RTLS) Abort Profile ........................ 22 RTLS Trajectory: Altitude vs. Time .............................. 33 RTLS Trajectory: Angle-of-Attack vs. Time ......................... 44 RTLS Flight Path Positions ................................... 55 SSME #1 Out Generic RTLS/PPA Trajectory Altitude Profile ............. 106 SSME #1 Out Generic RTLS/PPA Trajectory Angle-of-Attack Profile ........ 117 SSME #1 Out Generic RTLS/PPA Trajectory Mach Number Profile ........ 128 Correlation of Stagnation Heating Rate Plume Parameter with Radius ...... 149 Altitude-Velocity Profiles for PPA Maneuver ....................... 17
10 Heating Rate History for Aeroheating Trajectory .................... 1811 Heating Rate History for Plume Heating Trajectory ................... 19
List of Tables
1 Generic RTLS Plume Trajectory with SSME #1 Out (With Pitot Pressure) ..... 7
iv
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ViscosityStagnation pressureHeating rateSpecific heats at constant pressure and volume, respectivelyEffective radius
Stagnation heating rate plume parameterMach numberNusselt numberStanton number
Temperature"time
EnthalpyGas constant
Species mass fractionDensitySpeed of soundRatio of specific heatsEnergy per unit massDiffusion coefficient
Diffusion velocityEnthalpy per unit massReynolds numberSchmidt number
ContinuumFree molecularFree streamMixtureInternalTotal
Species i
Subscripts
Superscripts
Universal (gas constant)Normalized
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Section 1INTRODUCTION
If an SSME fails during the initial 160 seconds of flight, a return-to-launch-site (RTLS)maneuver will be implemented as shown in Fig. 1. The period of concern for this taskis the pitch-around maneuver where the vehicle is flying backward. Typical altitude andangle-of-attack histories during this portion of flight are shown in Figs. 2 and 3, respec-tively. Several distinct points have been identified during the turnaround maneuver.These points correspond to flight path directions on Fig. 4 where the estimated plumeboundary and shock location for position F, flying backward, are given. Thus, to definethe plume with a CFD analysis will require a large grid structure. At positions C and H,the free-stream flow along the plume boundary will produce a shear layer which mayimpinge on the ET and orbiter.
Three flight path positions were chosen for CFD analysis. Currently, the threepositions which were selected are:
Position C: High Mach number, possible shear layer impingement along plume bound-ary.
Position F: Flying backward, simplest applied case where plume is axisymmetric,similar to validation case.
Position H: Low altitude, low Mach number, possible shear layer impingement alongplume boundary.
The intent is to identify and define the flowfield at the most critical locations from anenvironment perspective. The boundary of the plume will generate a plume shock whichwill affect the airflow over the vehicle, hence, its stability. Plume/air mixture ratios will alsovary depending upon the location of the vehicle in its trajectory, and this has a stronginfluence on the heat load to the vehicle, specifically to the ET and Orbiter. Heatingdue to aerodynamic frictional heating is dominant if plume impingement is negligible.However, a large portion of the hot plume will be forced back to the vehicle due tothe vehicle's orientation against the oncoming free stream. Therefore, the most criticallocation from a heating concern is associated with the greatest plume impingement.The solution procedure used to predict the plume heating rates involves computationalanalysis (CFD) and engineering modeling (heating indicator approach).
Following this section, Section 2 details the trajectory used for the RTLS maneuverand the modeling (both CFD and engineering) needed to determine the flowfield. Section3 discusses the procedure for generating abort environments and provides the environ-ments for appropriate body points. Section 4 provided the conclusions reached by theThermal Panel and makes a set of useful recommendations for future work.
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Section 2MODELING FOR ABORT CASES
2.1 Trajectories
The trajectory used to study the intact-abort return-to-launch-site maneuver wasprovided by Rockwell [1]. For the abort trajectories, the SSME engine #1 out and #3 outcases were analyzed. The trajectories were based on statistically acceptable limits whichwould result in conditions to maximize plume heating. Only dispersed trajectories wereconsidered in the plume heating calculations. For the above abort cases, 1or dispersedtrajectories were used for an SSME #1 out case and an SSME #3 out case.
Table 1 provides the data for the plume trajectory with SSME #1 engine out (withpitot pressure). The power-pitch-around (PPA) portion of the trajectory is marked in thistable. The trajectory is plotted in Fig. 2. The details of the PPA portion of this trajectoryare given in Figs. 5 through 7, where altitude, angle-of-attack and Mach number historyprofiles are provided.
2.2 CFD Modeling
The following paragraphs describe the CFD study that was conducted to define thereturn-to-launch-site plume-induced thermal environment for the Space Shuttle. Thetask addressed herein is the CFD approach to model the plume flowfield during the PPA
maneuver. The motivation of the CFD study was to determine the flowfield in sufficientdetail to perform convective and radiative heating calculations.
First, it was necessary to develop a CFD procedure by validating the results againstmeasured data. The following assumptions were made to simplify the complexities ofthe flowfield:
1. No yaw was considered.2. The flowfields consisted of two frozen species mixing via convection only n one
frozen specie refers to the exhaust species and the other refers to air.3. Diffusion and chemical reactions were neglected.
The algorithm used for the Navier-Stokes solver consists of the following features:
1. Upgrade PARC2D to include the species continuity equation for two species with nodiffusion and chemical reaction.
2, Lag the species continuity equation behind the N-S equations.3. Couple the species boundary conditions implicitly to species continuity equation.4. Transfer information from the species continuity equation to the flow solver via mixture
gamma and transport properties.5. Use the original PARC2D solution as a restart to two-species computation.
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The second step to the CFD modeling was to use a composite plume for twoSSME plumes. Then the CFD methodology developed in the first step was used foran axisymmetric flowfield corresponding to location F in Fig. 4. This is the trajectorypoint where the angle-of-attack is 180 deg. In order to cover the whole range of angle-of-attack in the PPA maneuver, trajectory locations varying from C through H (Fig. 4)must be covered. At CFD points C and H, for example, a possibility of shear layerimpingement on the vehicle surface exists. Of these two locations, location C seems tobe more critical where the angle-of-attack is on the order of -135 deg, and there is apossibility of shear layer impingement on the tank base.
Only the first two steps of the problem were covered during this investigation. Thetwo technical notes resulting from the first two steps of this work are documented inappendices A and B.
2.3 Engineering Modeling (Heating Indicator Approach)
While conducting the CFD study, an easier method has to be sought to estimate,the upper level of heating on the ET during the power pitch-around maneuver of theRTLS. If a conservative method of prediction can be found which produces results notimpacting the ET TPS design, then flight assessments and design environments canbe produced in a simple manner for this extremely complex flow process. An earlyattempt was made by Rader [2] in the early 1970's at Northrop. This method has alsobeen utilized by Rockwell in a modified way. Both of these methods disregard most ofthe free-stream/plume interaction effects. The plume gas is assumed to be reversedand impinge at free-stream conditions on the ET components producing the worst caseheating. These heating rates are computed on a stagnation point.
Review of the method developed by Rader and used in part by Rockwell revealsinteresting results. The correlation due to Rader is essentially a different version of aFay-Riddell equation,
V Keff
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13
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Reff = 8 in., an extrapolation of the plotted data was performed up to R -- 10.0 ft. Theratio extrapolates to a value of 0.8 or higher, indicating that the equilibrium value isbeing reached for large radii.
In order to generate a new heating indicator procedure, a conservative philosophywas applied at REMTECH. The following assumptions were made in running a heatingindicator to an RTLS trajectory.
1. The flow considered chemistry effects.
Whether the flow chemistry is in equilibrium or the flow has finite reactionrates was considered by the code by turning on the appropriate flag in theinput. In order to check the chemistry of the flow, the equilibrium optionmust be turned on and the results compared against the finite rate results.The code used for doing these computations was BLIMPK [3].
2. The total enthalpy of the gas, Et, was assumed to be equal to total enthalpy ofthe plume gas.
In other words, no mixture of plume gas and air exists, thus giving themost conservative value for H_.
3. The total pressure of the gas, pt, was assumed to be post-shock total pressurebased on free-stream air.
No angle-of-attack effects were considered in the heating indicator cal-culations.
15
_ mct,,,_"l-_=" c H RTR 227-01
Section 3GENERATION OF ABORT ENVIRONMENTS
The total scope of this section should cover the environments of various body pointslocated on the ET. However, complete environments over the various body points arenot available now for reasons that will become clear later. The objective of generatingthese environments is to check any exceedances over design.
Work was initiated to compute the abort environments on the nose of the ET aftdome. This point may be the hottest of all other body points located in the rear of theET. The radius of the ET dome (=10 ft) was used to compute a heating indicator withthe assumptions made in Section 2.3. Two design trajectories, one for aeroheating andthe other for plume heating, were examined to compute the above heating. The altitude-velocity profiles near the PPA maneuver are given in Fig. 9. The heating rate history forthe aeroheating trajectory is given in Fig. 10. During the PPA period, the highest heatingoccurs around 530 sec with a heating rate of 2 BTU/ft2-sec. A similar set of calculationswas made for the plume heating trajectory. Figure 11 shows the heating rate history.Assuming equilibrium edge properties, both equilibrium and finite rate computations weremade using BLIMPK. The differences between these two computations are minimal.Both of these computations used 100 percent plume gas. When compared againstRI data book environments, the disparities are obvious. The RI peak heating is 0.9,as opposed to a value of 2.5 from the heating indicator. For comparison, the air-onlycomputations are also given in this figure. A number of discussions, teleconferencesand presentations have been held among REMTECH/MSFC, Lockheed/JSC and RI toresolve these differences. Although a lot of understanding of the methods from differentorganizations has developed, a definite consensus has yet to be reached as to the validityof the RI approach. Figure 11 shows results from Lockheed/JSC which are essentiallythe same as these from REMTECH. Another consideration was applied to REMTECH'scomputations. Since the PPA maneuver occurs in a relatively rarer atmosphere, it isworthwhile to check if the heating is transitional. Accordingly, a bridging relationship dueto Sherman was used in the following way:
Bridging Relationship: (Sherman)
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The data point was plotted in Fig. 11 showing that the transitional heating computationdid not bring the level of peak heating to the RI level.
Since the CFD computations for both 180 deg and 135 deg angle-of-attack have notbeen completed, it was not possible to compute either the convective or the radiativeenvironments. Although the 180 deg case has been reported in Appendix B, there seemsto be some sensitivity to the grid. As a result, this flowfield could not be used for heatingenvironments calculations.
20
F_E:rVl'rEC P--_ RTR 227--01
Section 4CONCLUSIONS AND RECOMMENDATIONS
4.1 Conclusions
The following conclusions have been made by the thermal panel after the numerousdiscussions and presentations:
1. The basis for RI/Downey's RTLS methodology is not clear.
2. A heating indicator approach alone will not be able to provide thermal environmentsfor design purposes.
3. However, a heating indicator can provide relative environments from one trajectoryto the next.
4. A suitable CFD study is required to define the flowfield for the complex interactionregion during a PPA maneuver.
5. The completed CFD studies both by REMTECH and other companies are not accu-rate enough to define the required thermal environments over the ET.
6. Several conclusions were drawn from the CFD experience of Rockwell/Huntsville andREMTECH in application to this problem:
a. The total flowfield is so large that initial gridding is difficult for proper capturingof the shock structure.
b. Gridding for the total flowfield makes gridding for detailed flow around the Shuttleconfiguration difficult.
c. The subsonic flowfield is quite large, making the placement of the outflow plumeboundary inaccurate if it is not placed far enough away from the mixing region.
4.2 Recommendations
Problems remain in validating the CFD flowfield for the Shuttle PPA maneuver. Sinceno wind tunnel or flight data are available, the CFD codes must be evaluated based uponthe physics of the flowfield and sensitivity to the grid structure. The required tasks tobe addressed are:
Task 1 u Grid Sensitivity Studies
1.1. Perform a detailed grid sensitivity study of 180 deg angle-of-attack case
1.2. Exercise the 3-D code for 135 deg angle-of-attack and provide details ofthe plume-air mixtures and thermodynamic properties of the flow near theOrbiter/ET base.
21
F:_ _" r,,,,,,1-r- _" C P..._ RTR 227--01
Task 2 m Physical Phenomena Validation
2.1. Use available data such as CALSPAN test data as benchmark cases to
evaluate the modeling of
a. Hot gas/air turbulent mixing, andb. Shock/shear layer interaction
Task 3 -- Document results and prepare a presentation of the results. Give salientfeatures, conclusions and recommendations and provide graphical data to
support each aspect of the problem.
22
F:_'r,,,,d-r"_'c _ RTR 227-01
Section 5REFERENCES
[1] Space Shuttle Generic ETR Plume Heating Data Book- External Tank u Ascent,Doc. No. SSD90D0016, Rockwell International, June 1990.
[2] Rader, R. J., "Stagnation Heating Rates in Rocket Engine Exhaust Plumes Consid-ering Equilibrium, Frozen, and Finite Rate Chemistry," Northrop TR-230-1186, Jan.1973.
[3] Murray, A. L., "Further Enhancements of the BLIMP Computer Code and User'sGuide," AFWAL-TR-88-3010, June 1988.
[4] Schaaf, S. A. and Chambr(}, P. L., F/ow of Rarefied Gases, No. 8, PrincetonUniversity Press, Princeton, NJ, 1961.
23
I=_'M-r"='C I--I RTR 227-01
Appendix ATwo-Species Variable Gamma Formulation
for a Forward-Firing Rocket Nozzle
24
I=_IE::r,,,.,,'l-I- _:: C I---.I RTN 227-O1
REMTECH TECHNICAL NOTE
SUBJECT:
DATE:
AUTHORS:
CONTRACT NO.:
PREPARED FOR:
Two-Species Variable Gamma Formulation for a Forward-FiringRocket Nozzle
October 8, 1990
Anthony J. Saladino and Sarat C. PraharaJ
NAS8-38185
George C. Marshall Space Flight Center
BACKGROUND
In the event of an SSME failure during the initial 160 seconds of the shuttle flight, a return-to-launch-site (RTLS) maneuver will be implemented. The period of concern is the powerpitch-around maneuver where the vehicle is flying backward at angles-of-attack close to180 deg. At these attitudes, a very complex flowfield involving severe free-stream/plumeinteraction affects results. This interaction includes mixing of a high Mach number hot
exhaust gas from the two SSME's with the low Mach number free-stream flow. It isbelieved that the hot gas is basically frozen in concentration levels and is allowed to mixwith air through convection and diffusion. The object of this technical note is to developan algorithm for an axisymmetric case to study the effects of variable "1'on the flowfieldof a two-species mixture. Since the species diffusion may not play a significant part inthe solution of the species continuity equation, it has been neglected in this analysis.
ANALYSIS
The PARC2DR [1] code was used as the foundation for the Navier-Stokes solver, andthe species continuity equation for mass fraction, c, of the exhaust species was coupledin an explicit fashion to the Navier-Stokes equations via the variable gamma. In twodimensions, the species continuity equation, after it was approximated by dropping theproduction term and neglecting the diffusion term, is written in physical space coordinates(_,=, y) as
Oc Oc OcN+u- m+vN =o. (1)Y
which is transformed to computational space coordinates (r, c, r/) as
Oc _. Oc v OCa--_"4- 0 _- 4- oq,r/ = 0 (2)
°
_z_"r,,._'r-_c H RTN 227-01
where r = t, _ = _C(z,y, t), 7/= 7/(z, y, t) and the derivatives are transformed by
U and V given by
+ qtc?_7
rlzC3v
Oy _vo_ + 77_0v
U = _t + u_=+ v_yV = qt +urlz +vrlv
(3)
(4)
are the contravariant velocities in the c and r/directions respectively, and _=, _y, r/= and
r/, are the metrics (_= = _, etc.) required for the transformation of the velocities, u andv, in the physical space (in the = and y directions, respectively) to computational space.The time derivatives of _ and 7/are zero since the grid is stationary.
Although the Navier-Stokes equations are solved in a steady state mode, the timederivative of the dependent variable is retained so that the resulting equations can beadvanced in time steps of Ar to a steady state, thereby allowing the solution to adjust toa converged condition over many iterations. Similarly, the species continuity equation (2)retains the time derivative and is solved in a steady-state fashion, where each solutionat time step n is used as the initial condition for the next time step n + 1; this processis repeated until the solution c and the solution vector for the Navier-Stokes equations[p, #u, pv, petot] do not change within a predetermined tolerance. Equation (2) can bewritten in discretized form as
c - c"+ + vo,,]c = o0 1
with 0f -- 0-"_ -/x-"_
0 1
and Or/= 0n - Aq
(5)
Rewriting Eq. (5) in a direction-split form (ADI, or alternating direction-implicit), results in
[I + A'r UO_][I + Av V077] cn+z = cn (6)
in which I is the unit matrix, and a term (Ar2UVO_Ov), which is a second order termin time, has been neglected. At small At, this term is much smaller than terms of theorder At. Therefore, Eq. (6) is the required form for the species continuity equation,and it is first order accurate in time.
In addition, upwinding is necessary so that communication between neighboring points
is physically correct. For example, if computations are being performed at grid point
(j, k), and the
U
• • •
j-l,k j,k j+l,k
contravariant velocity U is in the direction toward (.1",k) from (j- 1, k), then the derivativeof c in the direction of _ would be
c_ = cj,k - cj_l,k
Since ci, k is unknown and cj-l,k is a known value, upwinding also provides computationalstability. Covering all possible signs for c, upwinding for c in the _-direction becomes
and in the r/-direction,
{ cj,k--cj-t,k , U>OO_c= cJ+"_:_v-''_ , u=o
cj,k--cj+_,k , U<O
(7)
cj,k-cj,__z , V > Oa_c= %_+'T2%_-' , v = o (8)
Cj, k --Cj,k+ 1 , V < 0
Equations (7) and (8) are associated with the following form of Eq. (6), where absolutevalues have been taken of U and V:
[I + [z+ = d' (9)The solution of Eq. (9) involves solving each direction separately:
[Z + ATIVla,4c "+1 = "_" (10)
c ''+t = _' (11)
where
(12)
C = [z+ A,rlvla,7]-J _"
= [.[+ ,_-,-IVla,7]-t [I + _'lUla,,]-t,,(13)
l=_ =="r,,._ --r- _" _ k-I RTN 227-01
Inversion of the bracketed terms in Eqs. (12) and (13) is accomplished by first noting that
the resulting solution at point j, k will involve, at the condition U = 0 (say) points j + 1, kand j - 1, k, which corresponds to an equation of no more than 3 points. Therefore, atridiagonal solver is used to invert Eqs. (12) and (13). The value of Ar that is used inthe species continuity equation is kept equal to the Ar computed from the Navier-Stckesflow solver portion of the code. Using the same Ar allows the solution to progress in"time" at the same rate in both the flow solver and the mass fraction solver. Furthermore,
the tridiagonal inversion procedure incorporates the mass fraction boundary conditionsimplicitly to increase the accuracy of and reduce the convergence time of the speciescontinuity solver.
The reason for choosing an implicit, instead of an explicit, mass fraction solver is de-
scribed briefly below.
When considering the explicit mass fraction solver, only the interior flowfield points are
directly involved in the inversion procedure. The boundary points are imposed on theinterior points explicitly. For example, if a boundary condition is fixed (free stream) or free(supersonic outflow), these conditions are set at the points corresponding to the indicesassociated with those boundaries. After this step, the flowfield points are inverted and
the boundary points are supposed to feed back into the interior through derivatives.
¢/Boundary Conditions
:ilil::iiii_ili:i_ii:ii?i:iiii_ii_iiiii_ii_:iiiiiiiilili_ii_ii[ii:i!:!iiii:ili:i i:?i:i_ii_i!il:ii.ii_i!_i:_::ii:i::iiii:
i;i;i;i:;i':;i;i_RSIO_;
This is the same procedure which is used to invert the flow solver or Navier-Stokes (N-S) equations in PARC. However, whereas the interior points in the N-S solver transferinformation to the boundary points via the derivative term in velocity, there is no such
transfer from the interior [Eq. (9)] to the boundary for the mass fraction. For example,the tridiagonal terms in the matrix below are D, E and F:
E F 0
0
matrix for interior points in j-direction
4
I:_ ,_" t,,_ '-r" ,=="C I--I RTN 227-01
where E is along the diagonal and corresponds to the points (j, k), D corresponds topoints (3"- 1, k) and F corresponds to points (j + 1, k). Eq. (9) can be written for the./-direction only as
[iT -t- AT UO_] cn+l - cn (14)
so that if U is greater than zero, then Eq. 14 can be written as
( n+1_cn+1 "_ n+ u cj, j-l,k) =for a single value of j and k. Equation (15) can be rewritten in matrix form for differentvalues of j and constant k as
0 . r=(I+ArU)
(-A_u)(i+ A,u)
0 (-aTU) (1 -f- ATU)
where J goes from 2 to ira(j = I and J = J,,,,= are boundaries),
C2,1 ] n+l
C3,1
.(•Cjm,1
,Ira = jma= - 1
C2,1
C3,1
. CJm,1 .
(16)
(17)
andD = -ArU
E = 1 + ArU (18)
F=0
As Eq. (15) indicates for./= 2, values of c at location (1, k) would be required to evaluate
c2,_. However, location (1, k) is a boundary point and cannot be included in the vector ofc values in Eq. (16) since this is a matrix inversion for interior points only. Consequently,the boundary information for c is not being communicated to the interior points, resulting
in a non-changing value for c2,_, regardless of the initial values (for U < 0, the sameargument holds true at inflow boundaries; since there are boundaries in the forward-firing nozzle problem in which U < 0 coexists with U > 0, the final value of c will bezero everywhere, regardless of the initial values, due to the influence of both types ofboundaries on the interior point calculations).
Another way of describing the situation above is by making reference to the N-S equa-tions. The explicit solution of the N-S equations as programmed in the original PARCcode did not encounter such boundary communication problems for the solution to theinterior points. The corresponding vector c in Eq. (16) would be a vector of Q valuessuch that in a single direction,
[z+ O A]aQ"+1= RHS (19)
5
F=__=="rvl'T" _==="_ I---I RTN 227-01
where A is proportional to U. Since the N-S equations include terms related to the spatialderivative of velocity ((0_ U) Q,+z), the tridiagonal terms, for a boundary point at (j- 1, k)and interior points at (j, k) and (j + 1, k), are proportional to
E oc 1 + ( (2o)F _ Ar g(j+l,k)
The situation for the mass fraction equation is such that terms (U 0_ c"+l) result, leadingto
D,F c_ AT U(j,k)E oc 1 + ArU(j,k) (21)
where the U is now evaluated at point (j, k) only, so that there is no communication
from the boundary to the interior through U. Although derivative terms for c do appearin Eq. (9), these are part of the solution vector c"+l whose indices only include the
interior points.
Therefore, the implicit method was applied to the solution of the mass fraction equationby appending the boundary condition equation to the interior point Eq. (9). The form forthe boundary condition equations is different from those of the interior according to theboundary type m fixed (Dirichlet-type), isothermal (Neumann-type), etc.
Once the mass fraction c of the exhaust species is computed at all points, the ratio ofspecific heats 7 for the mixture is found as
cci_ -t- (1 -- c)c_ cp_ (22)
c c a + (1 - c)cv2 cv,,.
% and c_ are the specific heats at constant pressure and constant volume, respectively,for species 1 or 2, and subscript m denotes a mixture. Coupling between the speciescontinuity equation and the flow solver is accomplished through an equation of state ofthe form
P : (Tin- 1)peint (23)
where p is pressure, ,o is density and ei.t is internal energy, with 7,7, following fromEq. (22).
To accommodate the difference in the free-stream and exit nozzle value of gamma inthe normalization of PARC, the following changes had to be made.
Within the PARC code, the equation of state variables are normalized with respect tofree-stream conditions as
P P
-- p_a 2 -P p_(24)
T a
Too o, cc
6
i=_==" t',,._ -r- _-" c i..-i RTN 227-01
where the overbar denotes a non-dimensional quantity, a is the speed of sound, T is thetemperature, and oo corresponds to free stream.
In the calculation of the speed of sound or temperature, the updated code utilizes thenondimensional equations:
a 2 = 7raP , .E2 = 7m.___PP (25)P P
D
p=pRT _ T=7°°P_ (26)P
where the 3',,, in Eq. (25) was equal to 700 in the original code [1], since 3' was thesame at every point in the flow. For the present variable gamma approach, the constantGAMAREF was changed to GAMMA(J,K) (as computed from Eq. 22), with the referenceor constant (free-stream) value GAMAFS used in the calculation of temperature only andthe local value of GAMMA(J,K) used to determine the speed of sound.
One final consideration is needed to correctly account for the two-species variablegamma effects on the flowfield. This becomes important whenever the two specieshave different atomic weights. The equation of state is written in the usual form for a
single species as
where R = _, and ._
p = p}:tT (27)
is the universal gas constant. Eq. (27) can be normalized as
or,
or,
p ooa 2 = _ p oo-R.T Too
p=-_ T Too a
7oo
(28)
since'7oo .R T_ = a 2 (29)
For two species, R must be replaced by the gas constant for the mixture, R,n:
p = p RmT where,
R (30)
Rm'- l"_m
and M,, is the local mixture mass, so that upon normalizing Eq. (30),
.p = -_T R,n T_ (31)6oe 2
The mixture gas constant involves a summation over each species i,
Rm = E Ci Ri
p = _ T Tooac e 2 E ci -_.i
i
7_--R.00_ ci Ril
from Eq. (29)
Also,
Ri = cp_ - cv_
Substituting Eq. (34) into Eq. (33) results in
_T Eici(cp,- c_,,)p--_
700 Boo
(32)
(33)
(34)
_T cp,,- cv,,, from Eq. (22) (35)700 R00
Y (Tin - 1)cv,,,
'700 .R.00
Finally, the equation of state for two species can retain the form it had for a singlespecies by writing Eq. (35) as
_T= __ (36)
?
where700 R00
7 = (7.,- 1)c_,.A similar normalization of the first law of thermodynamics results in
(37)
P = (Tin - 1) p eint (38)
in nondimensional form. The temperature is then obtained from Eq. (36) in place ofEq. (26).
$
I=_t_M-r'_C H RTN 227-.-01
RESULTS
In order to validate the above algorithm, a few cases were examined. Results for aforward-firing sonic jet are shown for two cases below. Both cases have a ratio ofspecific heats equal to 1.382 in the external region of the flowfield and a free-streamMach number of 6.7. Case 1 uses a _ of 1.4 at the nozzle exit, and Case 2 uses a ..iof 1.3 at the nozzle exit. Case 1 corresponds to the experimental test case [2], whichhas nitrogen as the exhaust gas and a combusted mixture of methane and air as thefree-stream gas. In both cases the conditions in the free stream are
M_ = 6.7
poo = 0.3084 psia
Too = 332.53 ° R
Ttoo = 3318 ° R
Reoo/ft = 1.4 x 106
"Too = 1.382
and the total conditions in the sonic nozzle are
Tt = 495 ° R
pt = 44.206 psia
Ptj/Pt2 = 2.46
Since the O/F ratio of the nozzle exhaust was not known from any of the literatureavailable, evaluation of the exhaust and free-stream conditions was made in the following
manner. Determination of a mixture 7 at each flowfield point requires knowing c_, andc.,,,. Values of _ for the free stream and exhaust species were obtained from JANNAFtables, where air was assumed to be the free-stream fluid, and N2 was taken as theexhaust specie. The corresponding value of _ for air is 1.382, which was obtained from[3], and the exhaust _, was chosen to be either 1.4 (Case 1) or 1.3 (Case 2). The specificheat at constant volume was then calculated from the relation
cv_ = cp--t_, i = I, 2 (39)"/i
This procedure allows the computation of _,,,, to be made, but since the value of R,,, issensitive to the physically correct values of c_, and c,,,, _ will not be physically accurate(correct values of @, c,, and ..f can be determined for the exhaust and free-stream gaswhen the ODE (One-Dimensional-Equilibrium) computer code is run). Therefore, withinthe species solver, _was taken to be equal to _,,,,. Diffusion was assumed to be negligiblefor the blunt-body flowfield of the forward-firing nozzle. Inclusion of diffusion will beexercised at a later date to assess the importance of this term. A figure taken from Ref.
[2] is shown in Fig. 1 to clarify the nomenclature used in the following discussion.
Case 1
Contour plots of pressure, mass fraction of the exhaust species (c = 1 at the nozzleexhaust and c = 0 in the external region) and of the ratio of specific heats is shown inFigs. 3 through 5, where a plot of normalized pressure contours is given in Fig. 2 for thecase where 7= 1.4 everywhere for a single species model.
The value of gamma is equal to 1.4 everywhere in Fig. 2; in Fig. 3, gamma in the freestream is 1.382 and at the nozzle exit is 1.4. Comparison of Figs.2 and 3 indicateequivalent pressure contour values and shock locations. In addition, the mainstreambow shock is slightly closer to the nozzle for the case where 7 in the free stream is 1.382(Fig. 3) as compared to the case where 7 is 1.4 in the free stream (Fig. 2). Figures 4and 5 show similar trends in c and 7, respectively, in which the values at the nozzle exitchange smoothly to the free-stream values as one moves from the nozzle outward. Thelocation of maximum change in c and 7 occurs in a region normal to the nozzle axis, andis associated with the shear layer located between the mainstream bow shock and the jetnormal shock. Figure 6 is a side-by-side comparison of Mach number contours generatedby the variable-species option and the experimental results of [2]. The reattachmentshock starting behind the jet lip is clearly seen in the Mach number plot.
The next set of plots depicts the case in which 7, = 1.4, but includes a Wilke mixtureformulation for both viscosity # and thermal conductivity k to account for the effectsof a frozen mixture of two species. This is accomplished by utilizing a Sutherland
formula for each species i to compute individual transport properties #i and ki,. Thesevalues are combined along with the mass fraction ci into a mixture value for # and k.Figures 7 through 10 show no difference in the behavior of the flowfield as compared tothe Sutherland approximation for a single species as a function of temperature (Figs. 3through 6). Nevertheless, the Wilke formulation will be important for problems in whichdiffusion is not negligible and/or problems with more than two dominant species.
Case 2
In anticipation of a forward-firing SSME engine with an equivalent mass flow rate equal totwo SSME's, whose nozzle exhaust gamma is 1.3, the gamma for the nozzle in this studywas changed to 1.3 (while retaining a free-stream gamma of 1.382). Results for this caseappears in Figs. 11 to 14. Pressure contours (Fig. 11) indicate that the jet normal shockis somewhat farther from the nozzle exit as compared to the case where gamma at theexit was equal to 1.4. A calculation of p*/pt for a sonic nozzle (sonic divided by total)shows that the pressure is indeed larger at the nozzle exit when the value of gamma issmaller. The shock system between Case 1 and Case 2 is very similar. Exhaust massfraction contours (Fig. 12) range from 1.0 at the nozzle exit to 0.0 in the free streamwith a similar trend as in Case 1. The region of lower contour values near the nozzlecorner is due to recirculation which can be seen in the Mach number plot. This region ofreduced exhaust mass fraction occurs in the expansion fan emanating from the jet lip.
10
i=_z_-'r,,,_"l-_-"c H RTN 227-01
Evidently, lower value of c corresponding to the regions farther from the nozzle madetheir way to the expansion fan and did not reduce to zero as it did in Case 1.
Figure 13 indicates the gamma contours with 3' = 1.3 at the nozzle exit and 3' = 1.382 inthe free stream. The region of largest gradient is along the axis and corresponds to thejet-free-stream interface or shear layer. As the flow travels to the backside of the nozzle,the (convective) mixing region extends over a larger portion of the flowfield; the gradientsin the mass fraction and gamma are more gradual on account of the decreased velocity.A Mach contour plot (Fig. 14) describes the velocity field as in Case 1, where the jetnormal shock stands farther out from the nozzle exit with the lower 3'. The secondaryreattachment shock is clearly indicated in Fig. 14.
The final plot (Fig. 15) is a side-by-side comparison of the Mach number contours fromthe two-species variable gamma cases (with 3", = 1.4 and % = 1.3). This figure revealsthe difference in the locations of the respective jet normal shocks. Only the Sutherlandtransport property formulation for a single species is used. The lower gamma at theexhaust results in a jet normal shock that lies slightly farther from the nozzle exit.
REFERENCES
1. Uver, P. A. and Praharaj, S. C., "Upgrade of and Numerical Experiment with PARCfor Hypersonic Flow over a Hemisphere-Cylinder Configuration," REMTECH ReportRTR 181--01, Feb. 1989.
2. Nowak, Robert J., "Gas-Jet and Tangent-Slot Film Cooling Tests of a 12.5 ° Cone atMach Number of 6.7," NASA Technical Paper 2786, May 1988.
3. Macaraeg, M. G., "Numerical Models of Two Complex Hypersonic Flows," Journalof Thermophysics, Vol. 3, No. 2, April 1989, pp. 97-104.
t!
F=_t=='t,,,.,4-r-n_-c: H RTN 227-01
Mainstream bow shock--._
Jet expansion fan
Jet normal shock
radius, rn
Jet-mainstream
mixing region
Jet-mainstream
stagnation point
Jet-mainstreaminterface
Free-stream
expansion fan
Separated pocket
Secondary
Figure 1" Gas-Jet Shock Schematic (Adapted from Ref. 2)
.12
F=z_='-r'v't-r-_-c _ RTN 227-01
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F=__'- t,,,,_ --r" E_: C _----J RTN 227-01
M inf = 6.7Re/ft = 1.4E6
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Tt = 3318Rinf
Sonic Jet, T t j = 495 R, Ptj/Pt 2= 2.46
F_gure 6: Mach Contour Comparison Between Two-Species Variable Gamma Results (ExhaustGamma = 1.4. Free-Stream Gamma = 1.382) and Ref. 2
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e"XUJv
8
0U.
(_(1)
Cl,.O3
A
('-04""CO
0 •
oEO_o
•- LU
Q
Z II
U.
22
i=:z ,:" r,,,,,'t"r ,:- _ _-M RTN 227--01
0"0[
EE
¢/)"1
X
XLU
c:0
°_
014.
o3
°_
Q..O3
!
0
c-A
t.-_lOaO
',- o!.W,...o II
OEOEr- 0_.£ _U
•'--' U.O3
e- ,r,.X
ILl II
/.,i
LI-
Z3
I:=Z_" P,,.,,'I"T" I_: C I---I RTN 227-01
0'0
o.O
II
<1
\
\
X
II
EE
(.9
t-X
UJV
t-O
°_
EO
IJ-
°_
Q.
,5
"- II
,?:,E_
_EE_E_
-_U..
°_
u..
24
F::z_"rv't"r'_"_ _ RTN 227-01
oII
_._ 0_ 09
\\
X
A
O,Icoo3
II
EE
E
LI.
II
EE¢'I;I
or)
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LLI
e"O
°_
Et,.,
OI.I.
o_ 0_
25
F=z _=.- t,,,,,,I.-F- _- _ _..-! RTN 227.-01
Exhaust 'Y -- 1.4
Exhaust "/= 1.3
Figure 15: Mach Number Contour Comparison Between Exhaust Gamma Values of 1.4 and
1.3 (Free-Stream Gamma- 1.382)
26
I=_ E:I,v'I-I-I_:C P-I RTR 227-01
Appendix BVariable Gamma Computations for a
Forward-Firing Space Shuttle Main Engine
52
F=z_"r,v't-r'_'c H RTN 227-02
REMTECH TECHNICAL NOTE
SUBJECT: Variable Gamma Computations for a Forward-Firing Space ShuttleMain Engine
DATE: September 10, 1991
AUTHORS: Anthony J. Saladino and Sarat C. Praharaj
CONTRACT NO.: NAS8-38185
PREPARED FOR: NASAJMSFC
Abstract
A procedure for solving an axisymmetric SSME nozzle exhausting with an equivalentmass flow of two nozzles is presented for the return-to-launch-site (RTLS) maneuverfor an engine-out case. The algorithm incorporates a two-species variable gammaformulation along with diffusion terms associated with a species conservation equation.Coupling of this algorithm is accomplished explicitly to the Navier-Stokes PARC2DR andPARC3DR codes. Axisymmetric results indicate a very complicated flowfield with a bowshock extending to approximately 700 radii in front of the nozzle, and low temperatureson what would be an external tank (El') aft of the nozzle.
Background
In the event of an SSME failure during the initial 160 seconds of the shuttle flight, areturn-to-launch-site (RTLS) maneuver will be implemented. The period of concern is thepowered-pitch-around (PPA) maneuver where the vehicle is flying backward at angles-of-attack close to 180 deg. At these attitudes, a very complex flowfield involving severefree-stream/plume interaction effects results. This interaction includes mixing of a highMach number hot exhaust gas from the two SSME's with the low Mach number free-stream flow. It is believed that the hot gas is basically frozen in concentration levels andis allowed to mix with air through convection and diffusion. The object of this technicalnote is to develop an algorithm for an axisymmetdc case to study the effects of variable,,f on the flowfield of a two-species mixture.
Analysis
Species Solution Procedure
The PARC2DR [1] and PARC3DR codes were used as the foundation for the Navier-Stokes solver, and the mass fraction, c, obtained from the species continuity equation(with and without diffusion) was coupled in an explicit fashion to the Navier-Stokes
equations via the variable gamma and diffusion terms if diffusion is considered. In two orthree dimensions, the species continuity equation, after it was approximated by droppingthe production term, is written in physical space coordinates (t, x, y) as
0 pia--T + _ " (piV) = v. (pZh2 _7 _) (1)
where D12 is the binary diffusion coefficient, pi and c4 are the density and mass fraction ofspecies i respectively, and V is the velocity vector. Rewriting Eq. (1) using the definitionof mass fraction
results in
with _= _ _'+of continuity
P_ = ci (2)P
a(pc )+ _. (p Vc¢) = _7" (pD12 x7 _) (3)
j + _ Eq. (3) can be reduced to the following form with the use
Dci
P Dt =V'(PDI_ _c4) (4)
where the global continuity equation was used to simplify Eq. (4) and _ is the substantialderivative,
At this point Eq. (4) can be rewritten in a strong conservation form for three dimen-sions as
a(p,,_) O(pv_) a(p_,_)a (P_.___2)+ + + = _Z" (pV12 _ _) (5)
& O= Oy Oz
where u, v and w are the velocities in the z, y and z directions, respectively. EitherEq. (4) or Eq. (5) can be used to compute the mass fractions. The reason for using theweak form instead of the strong conservation form will be discussed next.
Upon normalizing the variables according to reference values
u v p z-_= P ,-_= _,_ = _,_= ,'_= m,y = .Y ,[= t a'ef (6)Pref aref aref "[refPref Iref lref lref
where a is the speed of sound, Eqs. (4) and (5) can be written as
o o oT + = + +
2
F:_==-r,,_'--i-_="c _ RTN 227-02
Reref -- Pref are! Iref#re!
E c -- ,up D12
The Schmidt number, Ec, will be taken as a variable quantity in this development.
To transform Eqs. (6) and (7) from physical space to computational space (G r/, _,r), start with the functional relationship between the coordinates: r =t, _ = _ (x, y, z, t),,7 = ,7 (=,y,z,t), _ = ¢ (=,y,z,t), so that the derivatives from physical to computationalspace become
8= "-"
Or+ _,0_+,, o, +¢, o¢_. 0_+.7=0, + ¢.o¢.q a_+,, 8, +¢y 8¢ (8)
and the contravariant velocities are defined by
U = 5+uf=+vfy+w_--
V = 7/t+uqz+wlu+w_
W = ¢t+u¢_+v_v+wG
(9)
where _= -- _, etc., are the metrics required for the transformation of the velocities u, v, win the physical space to the computational space. The time derivatives of _, r/and _ arezero since the grid is stationary.
The weak form of the species continuity equation in the computational space be-comes
I [ DI_ ) (10)cr + U c_ + V c_ + W c¢ = Rer,I V'_,#-_c V c
where the overbars and subscript, i, have been dropped for convenience, and thederivatives are given by Eq. (8). In this form, the species conservation equation canbe discretized and the left-hand side can be rewritten in direction-split or alternating-direction-implicit (ADI) form and the right-hand side is kept explicit:
[I + ArUO_][I + ArV3,][I + ArWO¢]c"+" = RILED (11)
where
RHED = cn + Ar
I is the unit matrix, 0,, = J- etc., and terms of second order in time have been
neglected. At small At, these second-order terms are much smaller than terms of orderAt. Equation (11) would be advanced in time using current values at time step n asinitial conditions for the next time step n+ 1.until steady-state convergence was achieved.
3
F:_ _" M-r" _" C _ RTN 227-02
Similarly, the strong form of the species conservation equation can be transformed tothe computational space after multiplying Eq. (7) by the inverse Jacobian j-l, resultingin the delta form
(13)
Rss=-zx, \ s )e \ J ]d
An additional term arising from this process
(14)
can be shown to be equal to zero [2].
The decision as to which of the diffusion equations to be solved u the weaklyconserved form of Eq. (11 ) or the strongly conserved form of Eq. (13) -- was made basedupon the type of numerical stability scheme desired. Solving the strong conservation formof the species equation would require either central differencing that initially includesartificial dissipation (since Eq. (13) is the same form as the fluid dynamic, centraldifference equations in PARC), or upwinding. It was decided that since the speciescontinuity equation can be quite easily upwinded, then Eq. (11) would be discretizedaccordingly.
One-sided differencing is utilized to bias the discretizing species continuity equationin the direction of the fluid flow; upwinding provides correct communication betweenneighboring points. For example, if computations are being performed at grid point (j, k),and the
U_
j- 1,k j,k j + 1,k
contravariant velocity U is in the direction toward (j, k) from (3"- 1, k), then the derivativeof c in the direction of _ would be
c_ = cj,k - cj-l,k
Since cj,], is unknown and cj-lj, is a known value, upwinding also provides computationalstability as it is naturally dissipative. Covering all possible signs for c, in two dimensions,upwinding for c in the _-direction becomes
[ ci,k--ci_l,k , U > O
O_c I c,+l,J,-ci-_,k U = 02
%], - ci+1,1, , U < O
(15)
4
)=:a_" rv't "T" _=" c v--¢ RTN 227-02
and in the )/-direction,
{ cy,s,-ci, k_l , V>Oi:9,1c= ci,k+,--ci,k-t V = 02 )
ci,k--c/,k+l , V<0
(16)
Equations (15) and (16) are associated with the following form of Eq. (11), where absolutevalues have been taken of U and V:
[z+ a,lulae][z+ a,ivla.]:+l = RHSD" (17)
The solution of Eq. (17) involves solving each direction separately:
[z+ zXrlVIO.]c"+_ = RHSD" (18)ft
c"+z = RHSD (19)
where
RHSD" = [I + A_-lUIO_]-': (20)
RHSD = [I + zx_-IVla.]-1 RHSD"(21)
= [z + _x_-IVla.]-'[z + _-lUlae]-z:
Inversion of the bracketed terms in Eqs. (20) and (21) is accomplished by first noting thatthe resulting solution at point j, k will involve, at the condition U = 0 (say) points j + 1, kand j - 1, k, which corresponds to an equation of no more than 3 points. Therefore, atridiagonal solver is used to invert Eqs. (20) and (21). The value of Ar that is used inthe species continuity equation is kept equal to the Ar computed from the Navier-Stokesflow solver portion of the code. Using the same Ar allows the solution to progress in"time" at the same rate in both the flow solver and the mass fraction solver. Furthermore,
the tridiagonal inversion procedure incorporates the mass fraction boundary conditionsimplicitly to increase the accuracy of and reduce the convergence time of the speciescontinuity solver. The reason for choosing an implicit, instead of an explicit, mass fractionsolver is described briefly below.
When considering the explicit mass fraction solver, only the interior flowfield pointsare directly involved in the inversion procedure. The boundary points are imposed onthe interior points explicitly. For example, if a boundary condition Is fixed (free stream)or free (supersonic outflow), these conditions are set at the points corresponding to theindices associated with those boundaries. After this step, the flowfield points are invertedand the boundary points are supposed to feed back information into the interior throughderivatives.
5
f:_ EE:r_-r"_:c _.l RTN 227-02
_" Boundary Conditions
!'!_!iiiiii_i,ii'_iiiii_ii!i:iii_ili i!_i!iii'i
This is the same procedure which is used to invert the flow solver or Navier-Stokes(N-S) equations in PARC. However, whereas the interior points in the N-S solver transferinformation to the boundary points via the derivative term in velocity, there is no suchtransfer from the interior [Eq. (17)] to the boundary for the mass fraction. For example,the tridiagonal terms in the matrix below are D, E and F:
"E F 0
0
matrix for interior points in j-direction
where E is along the diagonal and corresponds to the points (j, k), D corresponds topoints (j - 1, k) and F corresponds to points (j + 1,/¢). Equation (17) can be written forthe j-direction only as
[I + Ar UO_] c"+1 = RHSD" (22)
so that if U is greater than zero, then Eq. (22) can be written as
Cr_+l U/C n+l C r_'4"1_,k + A_- _ i,_ - j-l,k ) = RHSD'),k (23)
6
for a single value of j and k. Equation (23) can be rewritten in matrix form for differentvalues of j and constant k as
(1+ o
+ A.u)• n+l
c2,1
c3,1
• cJ'm,1 .
RHSD2,1
RHSD3,1
RHEDjm,1
where j goes from 2 to Jm (j = 1 and j = j,,,,= are boundaries),
J_ = jm,,= - 1
and
,, (24)
(25)
D = -ArU
E = 1 + ArU (26)
F=0
AS Eq. (23) indicates forj = 2, values of c at location (1, k) would be required to evaluatec2,k. However, location (1, k) is a boundary point and cannot be included in the vector ofc values in Eq. (24) since this is a matrix inversion for interior points only. Consequently,the boundary information for c is not being communicated to the interior points, resultingin a non-changing value for c2,_, regardless of the initial values (for U < 0, the sameargument holds true at inflow boundaries; since there are boundaries in the forward-firing nozzle problem in which U < 0 coexists with U > 0, the final value of c will bezero everywhere, regardless of the initial values, due to the influence of both types ofboundaries on the interior point calculations)•
Another way of describing the situation above is by making reference to the N-Sequations• The explicit solution of the N-S equations as programmed in the originalPARC code did not encounter such boundary communication problems for the solutionto the interior points. The corresponding vector c in Eq. (24) would be a vector of Qvalues such that in a single direction,
[I + Ar O_A] AQ _'+1 = RHS (27)
where A is proportional to U. Since the N-S equations include terms related to the spatial
derivative of velocity ((0_ U) Q,,+I), the tridiagonal terms, for a boundary point at (j - 1, k)
7
F:_=='r,,,,,d'-r-=="c _ RTN 227--02
and interior points at O',k) and (j + 1, k), are proportional to
Doc Ar U(j - l,k)E _ l+_rU(j,k)F c_ Ar U(j+t,k)
(28)
The situation for the mass fraction equation is such that terms (U oq_c"+1) result, leadingto
D,F o¢ Ar U(j,k) (29)E oc 1+ z&rU(j,k)
where the U is now evaluated at point 0", k) only, so that there is no communicationfrom the boundary to the interior through U. Although derivative terms for c do appearin Eq. (17), these are part of the solution vector c"+1 whose indices only include the
interior points.
Therefore, the implicit method was applied to the solution of the mass fractionequation by appending the boundary condition equation to the interior point Eq. (17). Theform for the boundary condition equations is different from those of the interior accordingto the boundary type -- fixed (Dirichlet-type), isothermal (Neumann-type), etc. Along asolid boundary the mass fraction gradient is set to zero, while along a free boundary themass fraction is either extrapolated from the interior flow as in the subsonic case or setequal to the value from the preceding grid point as in the supersonic case.
Energy,,.Equation
The contribution of diffusion to the energy equation appears in the heat flux term(neglecting the Dufour Effect)
q_ =PEhic4 f_ (30)i
where h_ is the enthaipy of component i, and the mass diffusion velocity _4 is relatedto Fick's Law of Diffusion [3] as
-Dq= _ (31)
c-4
Normalizing Eq. (30) results in the following term in physical space
\s v (32)
where the Schmidt number, Sc, is defined by
So= # (33)pDii
The Schmidt number is allowed to vary in this formulation.
8
i:_ ==" I',,,,'1"1- IE: C I-..-I RTN 227-02
The diffusion coefficient was computed from the following equation obtained from thebifurcation approximation of Bird [4]
D
Dq = FiFj (34)
where _ = l"rzg×zo-'(T)l"659cm2/sec with p in atmospheres and T in degrees Kelvin,pand Fi = (rni/26.7) °'4s9, where mi is i th the species molecular weight. In discretizingDz2, it is advisable to smooth out the significant nonlinear variations in its value acrossadjacent cells. This reasoning, plus the differences in the cell sizes, suggests an averagediffusion coefficient [5] which guarantees the flux entering an interface from the left of adiscretized cell is equal to the flux exiting that interface at the right:
Di Di+i(Xi+i- Xi)
where the species subscript 12 has been removed as it is understood that D is the binarydiffusion coefficient, and where / + 1 refers to the actual computational grid center and
i + ½ indicates a cell interface:
i,j+ l
i - 1,j I i,j !o I • I o
I I
i- [,j ---'_L2
+½,;
Additional Considerations
Discretization of the species second derivative terms in computational space isaccomplished according to whether the derivative is mixed or not mixed. For similarsecond-order derivatives, second-order central spatial difference operators are used;whereas, when applying a mixed derivative to c, average derivative values are computedat locations between the two grid points in a particular derivative direction.
O_ C_ = (Cj+l,k,l - Cj,k,l) -- (Cj,k,| - Cj-l,k,')
69, C.n --" _1[ cj+l'k÷l'l-cj'l'l'k-l'12 4- CJ'k-I'l'l--Cj'k-l'l]2 (36)
1 [Cj,k+l,l--Cj,k-l,t + Cj-l,k+l,l- C]-l,k+l,l ]2 2 2
9
F:;Z==='Pv't"F'==='C _ RTN 227-02
Since the implicit equations are solved or inverted for the c,. along a single 3"or k or I linewhile changing one of the other indices (as opposed to solving for the species for theentire computational space in a single inversion), provisions were made to incorporatecyclic iteration between the possible sweep direction ordering.
Finally, to provide for axisymmetric diffusion, the species continuity
(37)
in which v is the radial velocity and y is the radial coordinate, was transformed tocomputational space coordinates and coded in the upgraded PARC code, hereafterreferred to as PARCSPEC.
Species/Flow Solver Coupling
Coupling of the results from the species concentration solver to the fluid dynamicsolver is accomplished in an explicit fashion. The continuity and momentum equationsremained unchanged in form; however, due to the use of Wilke's mixture formula tocalculate the transport properties, the energy equation changes its form to incorporatea variable thermal conductivity and cp. The mass fractions, along with the species gasconstants, are used to compute `7 at every point in the flowfield, which replaces theconstant `7 required in the original version of PARC. The ratio of specific heats `7 for themixture is found as
c_, + (1 - c)c_ _ 2.,"7,..= - -- (38)
c c,, + (1 - c)c,, c,..
cp and c.,, are the specific heats at constant pressure and constant volume, respectively,for species 1 or 2, and subscript m denotes a mixture. Coupling between the speciescontinuity equation and the flow solver is accomplished through an equation of state ofthe form
p = (`7,n - 1)pei,t (39)
where p is pressure, p is density and e{,t is internal energy, with `7,,, following fromEq. (38).
To accommodate the difference in the free-stream and exit nozzle value of gammain the normalization of PARC, the following changes had to be made.
Within the PARC code, the equation of state variables are normalized with respectto free-stream conditions as indicated in Eq. (6) plus the additional terms
T aT = -- _ = m (40)
where T is the temperature, and a is the speed of sound.
In the calculation of the speed of sound or temperature, the updated code utilizesthe nondimensional equations:
a2 = `7,','=P , _2 = `7,',,____PP (41)
10
i:_ EE r,,,A"l" ==" c _ RTN 227-02
P=PAT _ _=Tr,/P_ (42)P
where the 7,. in Eq. (41) was equal to 7oo in the original code [1], since 7was thesame at every point in the flow. For the present variable gamma approach, the constant
GAMAREF was changed to GAMMA(J,K) (as computed from Eq. 38), with the reference
or constant (free-stream) value GAMAFS used in the calculation of temperature only andthe local value of GAMMA(J,K) used to determine the speed of sound.
One final consideration is needed to correctly account for the two-species variable
gamma effects on the flowfield. This becomes important whenever the two species havedifferent molecular weights. The equation of state is written in the usual form for a single
species as
p = pRT (43)
where B = _, and._ is the universal gas constant. Equation (43) can be normalized as
2a,,/= -#p,,/R Tr s
-- TrefRor, _ =_T 2
aref
or, _ ="Yre f
(44)
since
7,_/ R T,=/ 2--" are f (45)
For two species, R must be replaced by the gas constant for the mixture, P,_:
p = p P_T
R,m = _Tl'tm
where,
and m,_ is the local mixture mass, so that upon normalizing Eq. (46),
= R,. T,,j,2
aref
The mixture gas constant involves a summation over each species i,
R,_= E c_R_
(46)
(47)
(48)
11
t=_'t,,/l"T'_'C H RTN 227-02
Als0,
_-Too
I
_- c,.R,
i
/74 = %_ - c_
Substituting Eq. (50) into Eq. (49) results in
_ _(cp, - _,)_T /
p-
from Eq. (45)
(49)
(50)
N
T %. - c_., from Eq. (38) (51)_ref Rref
_T--" m
%ef
Finally, the equation of state forspecies by writing Eq. (51) as
(-y,,, - 1)c,,,
two species can retain the form it had for a single
w
_T= -- (52)
where"fooP_o
"r= (._,._ z)_..A similar normalization of the first law of thermodynamics results in
= (-_,,,- 1) : ei.tin nondimensional form.
Eq. (42).
(53)
(54)The temperature is then obtained from Eq. (52) in place of
Results
in order to validate the above algorithm, a few cases were examined. Results for a
forward-firing sonic jet [6] for two cases _f_, = 1.4 and *f,v = 1.3 at a .,/fs = 1.4 comparedvery well with experimental results. Diffusion was not invoked in the code in that study.
The engine-out condition on a PPA maneuver in an RTLS trajectory was modeledas a single nozzle with a mass flux equal to twice that of one SSME, and with a radiusscaled to V'2 times a single SSME radius. A grid was created with INGRID [7] assumingthe nozzle to be a simple outflow plane, with a straight afterbody with a radius of 1.05 r;the location of flow impingement on the external tank (ET) (which is not part of the gridmodel) would then be approximated by examining the solution in the region where theET would exist in relation to the SSME's.
12
I:_ _" r,,,,'t -I" _" C I---,I RTN 227-02
Conditions on the nozzle and free stream are as follows:
Table 1: Nozzle and Free-stream Conditions
p (Ibm/ft3)
T (°R)
p (Ibf/in2)
m (Ibm)Mach Number M
'7
Nozzle Free Stream
3.226 x 10 -3 9.0328 x 10 -6
2193.0 409.18
2.619 1.3681 x 10 -3
14.132 28.858
-4.67 1.464
1.20 1.40
The free-stream conditions were taken as the reference conditions in PARCSPEC.
This study focuses on the possible impact on the thermal environment caused bythe SSME firing in the same direction as the Shuttle flight direction. At a flight angleof 180 deg, the plume impingement on the ET may not occur; however, at a flight angleof 135 deg, the free-stream air may force back the plume which is expanding from thenozzle into a shear layer that impinges the ET.
Axisymmetric SSME
The extent of the grid ranges from 1454 (two-nozzle equivalent) radii in front of thenozzle and 300 radii behind the nozzle, exit plane along the axis, and 2800 radii inthe radial direction. A grid size of 140 points (axially) and 340 points (radially) wasconstructed with clustering near the axis and nozzle expansion region (Fig. 1). Thenozzle is located at the = = 0 location on the figure. The large size of the grid preventsthe nozzle from being seen on this scale. Due to the very large physical extent of thisgrid, the grid lines are highly compacted in the vicinity of the nozzle, and more sparse asthe grid extends far away from the body. Smoothness between adjacent grid cells wasadhered to as best as possible to limit the nonlinearities associated with computationsacross such cells. Diffusion was neglected in this set of runs.
Convergence was difficult to obtain when the conditions specified in Table 1 wereimposed at once. Instead, the nozzle conditions were taken at their specified values,except that "r_" was set to 1.4, and the free-stream conditions were adjusted graduallybetween restart files until the specified conditions were reached. Each set of free-stream
conditions was run to a reasonable convergence before adjusting the conditions for thenext case.
Contour plots of streamlines are shown in Figs. 2-3 for the case where "(t," = -7fs= 1.4 and the molecular weight n=,,,,= rn.fs = 28.858. The demarcations on the axisspecify a range of 200 radii. A smooth variation in the streamlines is indicated in thesefigures, where the flow turns back at a point equivalent to 180 radii in front of the nozzleon the axis.
13
_=_:rvl"T'==='C P-I RTN 227-02
The next step in the convergence process was to reduce _'_r and m,v from 1.4 and28.858, respectively, to 1.2 and 14.132, which corresponds to the H2 - 02 system value.Results for the PPA conditions at c_ = 180 deg are presented in Figs. 4 - 8. The bowshock is located at roughly 700 radii in front of the nozzle while the nozzle shock occurs at250 radii before the nozzle (Fig. 4). Due to the high nozzle pressure and low atmosphericconditions, the nozzle shock extends a great distance away from the nozzle and is highlycurved as a large expansion from the axis to the free-stream condition must occur. Thevalues of gamma (Fig. 5) indicate a smooth variation in the extent of the mixing region.Along the axis, supersonic Mach numbers of Fig. 6 show the increase of the value of4.87 at the nozzle to 10.9 about 170 radii before the nozzle and then a decrease to
zero at an X/R value of 200 where the two opposing flows stagnate. The behavior ofthe streamlines (Fig. 7) in front of the nozzle is due to the small gamma and molecularweight of the exhaust gas compared to that shown in Fig. 2. Clearly, the hot plume gasdoes not impinge on what would be the ET. A close view of the nozzle exit temperatureprofiles (Fig. 8) indeed verifies the low temperature in the region just behind the exhaust.In fact, the temperature is close to the free-stream value.
Results In Progress
When the nozzle gamma was changed from 1.4 to 1.2, it was found that thestreamlines fluctuated over a large distance along the outflow plane. This is reasonedto be caused by flowfield and exit plane coupling which could be eliminated if the exitplane were extended farther downstream. An extension of 400 radii m or a total of 700radii -- comprise this new grid, and results are being generated at the current time.
In addition, the diffusion upgrade to the two species methodology is being checkedout for the two-dimensional sonic jet to ascertain the effect that diffusion has on thevariable gamma solution.
To study the three-dimensional effects on the RTLS flowfield, the axisymmetdc gridwas revolved about the axis and points between the outer boundary and the bow shockwere eliminated, as these points were originally employed to capture the unknownlocation of the shock. Eighteen planes comprise a total space of 180 deg with halfof the domain being a mirror image of the constructed grid domain. Computations arein progress for this case.
References
1. Uver, P. A. and Praharaj, S. C., "Upgrade of and Numerical Experiment with PARCfor Hypersonic Flow over a Hemisphere-Cylinder Configuration," REMTECH ReportRTR 181---01, Feb. 1989.
2. Pulliam, T. H., "Euler and Thin Layer Navier-Stokes Codes: ARC2D, ARC3D," Notesfor Computational Fluid Dynamics User's Workshop, The University of TennesseeSpace Institute, Tullahoma, TN, March 12-16, 1984.
3. Kuo, K. K., Principles of Combustion, John Wiley and Sons, 1986, p. 638.
14
F=z_" M-r" _=" C _ RTN 227-02
4. Evans, R. M., "Boundary Layer Integral Matrix Procedure," Aerotherm Report UM-75--64, Aerotherm Division/Acurex Corporation, Mountain View, California, July 1975.
5. Oran, E. S. and Borris, J. P., NumericalSimulation of Reactive Flow, Elsevier SciencePublishing Co., Inc., 1987, pp. 237-240.
6. Saladino, A. J. and Praharaj S. C., "Two-Species Variable Gamma Formulationfor a Forward-Firing Rocket Nozzle," AIAA Paper 91-1349, 26th ThermophysicsConference, June 24-26, 1991, Honolulu, HI.
7. Dorrell, E. W., Jr. and Soni, B. K., "INGRID: Interactive Two-Dimensional Grid
Generation," AEDC-TR-86-49, Arnold Engineering Development Center, Tullahoma,TN, Feb. 1987.
15
i=_ ¢=- I,,/1 -i-. _- C I__..I RTN 227-02
0
000
0
0O-
C}
60-0
-600.0
X/R-300.0 300.0
Figure 1" Grid
16
0,,¢
I!
I!
p-
0
cO¢)c--
E
<Dk,.
o_
LI.
17
O
m,
RTN 227--02
C)It)
rr
0',¢;1"
I,,=.
II
II
0
(/)X<
¢.-O
<
e-o....
E
/6
-.,i.__
q m.O
¢,r
O
O
18
,=z,=-r,,,,d'-r',="c I--4 RTN 227-02
0
0
00
0
0_r
-I000.0
I I|
-700.0 -400.0 - 100.0 200.0
Figure 4: Pressure Contours for 7,v = 1.20, 7fs = 1.40
19
I=_ IE r,_-r" IE <_ I--I RTN 227-02
O
O
O
0
0
oO
0
00
0
0I
-I000.0 -700.0
A 7 = 0.01
l 1.2
-400.0 -I00.0 200.0
Figure 5: Gamma Contours for "f,v : 120, o,S.s : 1.40
2O
I=_ I_: r,v'l-I- _: C !--4 RTN 227-02
O
(5OOC_
O
OO¢D
C}
OOOJ
O
OO¢0
O
c_O,q,
AM= 1.0
0
C), I i
-1000.0 -?00.0 -,100.0 -100.0 200 0
Figure 6: Supersonic Mach Contours for _',v = 120, 1,fs = 1.40
21
i=Zl_:r,,,,,,l"l-lE:C P...i RTN 227-02
O
OO-O
0
00a3
-I000.0
0
200.0
Figure 7: Streamlines for 7iv= 120, 1,is : 1.40
22
I:_ ==" I,,,,,1 "1- E: C I--I RTN 227-02
O
C)
TT--_
I
-t0.O -5.0 0.0
x/t .
5.LI
Figure 8: Temperature Contours for ")'x = 120, -),is = 1.40
23