I)',' 

IMPROVED TWO-DIMENSIONAL-KINETICS 

C/( 5r/939* 

COMPUTER PROGRAM 1- 

NAS8-35931 ' 

Second Quarterly Progress Report 

1 April 1984 through 30 June 1984 

/ 71/07 

(NASA-CR-q=&%W INPhCVED 

2-DIMENSIONAL-KINETICS CONEUTEB FBOGRAH 

Quarterly Progress BeFcrt, I Epr. - 30 Jun. 

1584 (Software and Engineering Associates, 

Inc.) 36 p HC AC3/NF A01 CSCL 09B G3/61 

Unclas 

43222 

Prepared by: Gary R. Nickerson 

Software and Engineering Associates, Inc. 

1560 Brookhollow Drive, Suite 203 

Santa Ana, California 92705 
IMPROVED TWO-DIMENSIONAL-KINETICS COMPUTER PROGRAM 

1 .O ------ BACKGROUND 

Future Orbit Transfer Vehicles (OTV) presently under 

consideration need rocket engines delivering a high specific 

impulse. This high performance can be obtained with large 

area ratio thrust chambers using oxygen with either hydrogen 

or hydrocarbon fuels. In the projected nozzles the 

combustion products are expanded to low pressure and 

temperature levels at high Mach-number flow, a domain which 

has not been experienced with existing rocket engines. 

Some modifications have recently been incorporated in 

the Two-Dimensional-Kinetics (TDK) computer program, such 

as: Condensed phase flow simulation, treatment of shock 

waves induced by the wall curvature, and the coupling of the 

TDK code with the Boundry Layer Module (BLM) for an improved 

thrust chamber specific impulse prediction. 

Recent boundary layer calculations for large area ratio 

nozzles have revealed that the boundary layer becomes very 

thick in the region of high Mach-numbers. This result is 

quite different from rocket engines built up to now and 

requires an examination of the existing thrust loss 

calculation method due to the viscous effects adjacent to 

the wall. Furthermore, the knowledge of the Knudsen-number 

is desirable which differentiates between continuum, slip 

and free molecular flow, and thereby identified when the 

. .-.ahalysis, presently based on Newtonian fluid flow, becomes 

questionable or invalid. 

.. 

1 
In order to obtain highly accurate performance 

predictions, the best supporting data for equilibrium and 

finite rate chemistry must be prepared. Furthermore, the 

large area ratio nozzles require many more calculation steps 

to cover the complete nozzle flow field. This increases 

computation time significantly and favors an error 

accumulation which should be reduced as much as possible. 

Another point of importance connected with the OTV is the 

existence of ionized combustion products which permits 

vehicle tracing or interferes with the transfer Of 

communication or command signals. 

The current effort of shock wave modeling induced by 

the curvature of the wall should be expanded to permit the 

shock simulation caused by wall contour discontinuities. 

Also the Mach shock existence in the center of the flow 

field interacting with other shocks as well as the 

analytical treatment of multiple shock waves may 

significantly affect the interior nozzle flow and associated 

performance and need to be simulated. 

2 
2.0 OBJECTIVE 0 

The objective of this effort is to increase the 

analytical capability of the existing TDK/BLM computer 

program for performance of OTV thrust chambers. Areas which 

need further examination and improvement are the thick 

boundary layer inviscid core flow interaction and the 

related thrust loss calculation. To warrant highly accurate 

results the provision of the best available program input 

data is mandatory as well as the use of sophisticated 

modeling techniques to reduce computation time with advanced 

error control criteria. The simulation of wall shocks, 

Mach-shocks in addition to shocks induced by large Concdve 

wall curvature is essential since this interaction with each 

other produces flow field changes which in turn affect the 

nozzle performance. 

.. 

3 
The program plan for this work is presented in Figure 

1. Progress in each of the work tasks is presented in the 

sub-paragraphs 3.1 through 3.5. One trip was also taken 

during the reporting period. 

Mr. Nickerson and Dr. Dang spent Wednesday, 23 May 

1984, at NASA MSFC discussing the status of the contract 

with Mr. Gross and Mr. Krebsback. The 24th of May was 

spent in attendance at the Performance Standardization 

Subcommittee Liquid Panel Meeting held at NASA MSFC where 

Mr. Nickerson gave a technical presentation on the work 

being performed under this contract. 

Work has continued on this task throughout the 

reporting period. A report draft has been prepared on this 

subject for SEA by Dr. Waldman and is attached. 

3.2 CHEMISTRY IMPROVEMENTS 

The ODK and TDK modules have been modified to treat 

electrons and ionic species. To do this required 

establishment of a proper linkage of these modules with the 

ODE module. This task, 3.2h, was requested by the AFRPL. It 

is particularly useful for obtaining nozzle exit plane 

properites to be used as starting conditions for exhaust 

plume calculations. Reference 1 is a good source for 

reactions and rates for these processes. 

-- 

Ref. 1: Jensen, P.E. and Jones, G.A., ttReaction Rate 

Coefficients for Flame Calculations" Combustion and 

Flame 32, 1-34 (1978) 

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ORlGlNAE PAGE 88 

OF POOR QUALITY 4 

5 
3.3 -_INOZZLE SHOCK --- WAVES 

ORIGINAL PAGE 19 

OF POOR QUALITY 

The remaining shock wave task to be performed is Task 

3.3c, multiple shocks. A detailed approach to this problem 

has been prepared and was presented to the COR for his 

approval at MSFC on 28 May, 1984. A copy of the plan, which 

consists of four tasks, is attached. Tasks 1) and 2) have 

been done. We are awaiting approval on Tasks 3) and 4). 

The SSME was used as a teat case in performing.Tasks 1) 

and 2). This is an engine of primary interest to NASA/MSFC. 

Because of the presents of Induced shocks in the nozzle, and 

also due to the near equilibrium thermodynamic state of the 

exhaust, the SSME is a very demanding teat case. For the 

right-running characteristics construction method, several 

numerical repinements were found necessary. These had to do 

with point insertion and the method for maintaining mass 

flow balance. The SSME was successfully run and the TDK 

program modifications were transmitted to NASA. 

3.4 A STUDY OF THE BLM WALL HEAT -----I_- FLUX OPTION 

The BLM contains options through which either the wall 

temperature, or the wall heat flux, can be specified. Since 

measurements of wall heat flux are sometimes available, the 

usage of these as a boundary condition for BLM was 

investigated. 

Since an input wall temperature profile will produce an 

output wall heat flux profile, computer runs were made to 

determine 1) the consistency, and 2) the sensitivity Of the 

computations. The ASE 400:l engine was used as a teat case. 

. .. First, zero wall heat rlux was requested and the adiabatic 

wall temperature computed. On a second run, the computed 

adiabatic wall temperature was input and it was shown that 

Zero heat flux was computed. Next a similar pair Of runs 

were made with a 2000O R wall temperature. This is much 

cooler than the adiabatic wall. Again, the results checked. 

6 
ORIGI'NAL PAGE 19 

OF POOR QUALlrV e 

This was followed by a pair of runs using a wall 

temperature profile provided by NASA, Figure 2, that 

includes the effects of regen cooling. The resultant wall 

heat flux is shown in Figure 3. Input of this heat flux 

then produced the wall temperature profile shown in Figure 

4. The predicted temperatures follow those of Figure 1, but 

show sensitivity because the influence 

is evidently large. 

coefficient aTw/agw 

To obtain a measure of this sensitivity, the profile 

for the heat flux out of the wall was decreased by 5% and the 

case was again run. The result Is shown in Figure 5. Wall 

temperatures were doubled, indicating that aTw/2Gw -20. 

The shape of the Tu profile is unchanged. The conclusions are 

that: 

1) 

2) 

The Tw and 4, computations are consistent. 

Local 4" is governed by local Tu, and visa versa. 

3) Tu Is highly sensitive to Q,, 

6, Is unsensitlve to Tu, 

4) Except for the adiabattc wall case, Bw - 0, the and 

input of Gw is not very useful. Wall temperature 

can not be estimated by input of a finite heat 

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4.0 ------ WORK PLANNED FOR THE NEXT --------------- REPORTING PERIOD 

The program plan shown in Figure 1 and during the next 

quarter are discussed below. 

The evaluation of high Mach number and low density 

effects on the existing boundary layer thrust calculation is to 

be completed. 

The low temperature data task, 3.2e, will be done 

early, probably during July. 

The major task to be worked will be multiple shocks, 

task 3.3~. 

.12 
5.0 ----------I---- 

CURRENT PROBLEM AREAS 

The problem areas discussed in the First Quarterly 

Report have been resolved. Our difficulties with invoicing the 

work was overcome when a first payment was received at the end 

of May. 

Thus, the first five months of the contract were 

financed by SEA. This had an adverse effect in that the level of 

effort was reduced so that the company cash flow could be 

maintained. At this time the work is about one month behind 

schedule. 

6.0 --- COST STATUS 

As of the end of the reporting period, the cost status 

of this contract is as follows: 

1) Total Cumulative Costs through 

30 June 1984 

8 42,255 

2) Estimated Cost to Complete 8 76,605 

3) Estimated Percentage of Physical $ 35% 

Completion of the Contract 

The costs on the contract to date correspond well with 

the estimated percentage of completion. 

13 
OTV Nozzle Design 

Thick Turbulent Boundary Layer 

18 June 1984 

Prepared for 

Software & Engineering Associates, Inc. 

1560 Brookhollow Drive, Suite 203 

Santa Ana, California 92705 

Prepared by 

Cye H. Waldman, Ph.D. 

Consultant 

5 15 Hermes Avenue 

Leucadia, CA 92024 

(6 191 942-30 16 
OTY Nozzle Design 

Thick Turbulent Boundary Layer 

Page 1 

1. Statement of the problem 

The JANNAF rocket thrust chamber performance prediction procedure assumes 

that the nozzle flow field can be separated into a boundary-layer region near 

the wall in which viscous effects are predominant, and an invisicd core flow 

containing the bulk of the nozzle mass flow. Basically, the procedure consists 

of calculating the inviscid flow within the nozzle (the so-called core flow), 

and then using the flow conditions found adjacent to the wall as representing 

the inviscid edge of the bOund8ry layer. Conventional methods, such as 

embodied in the BLIMP-J[l], TBLl21, or BLM[3] computer programs, are then used 

to compute a displaced wall, i.e., the wall position is displaced by the boundary 

layer displacement thickness. The process is iterated, if necessary, and a 

value for the thrust deficit is computed. 

However, nozzle designs for OTV applications can be such that most of the 

nozzle mass flow is confined to 8 region that is relatively near the wall. This 

is because of (a) compression of the exhaust gas caused by inward turning of 

the nozzle contour and (b) area increases significantly near the nozzle wall 

(?-effect). Under these circumstances the boundary layer can grow to such an 

extent that it interacts with the core flow. 

The effect of a thick boundary layer may be significant on the thrust deficit 

calculation and the JANNAF procedure must be re-examined. There are three 

effects of potential importance under these conditions which are not accounted 

for in the classical boundary layer theory: these are the effects'of 

0 transverse curvature 

longitudinal curvature 

8 fluid stratification 

In the following sections these effects will be examined in the light of their 

impact on the mean flow and eddy-viscosity models. Following that, 

recommendations will be made regarding improved thrust deficit calculstions. 
OTV Nozzle Design 

Thick Turbulent Boundary Layer 

Page 2 0 

2. Transverse Curvature Effects 

Transverse (or lateral) curvature effects become important in axisymmetric 

flows when the body radius is of the same order as the boundary-layer 

thickness. The transverse curvature term falls out naturally in the 

axisymmetic boundary layer equations when transforming from cylindrical to 

'body coordinates (see, for example, Cebeci & Bradshawt41). The transverse 

curvature term is included in the BLM program of CebeciI31. 

In turbulent flow, the effect of transverse curvature is to alter the mixing 

length. For external flows it is well-established that mixing length is smaller 

than its two-dimensional equivalent. Physically, the cylinder has less ability 

to create turbulence than a plane surface. This has been amply demonstrated in 

the literature; many experiments have been conducted on the effect of 

transverse curvature on thick turbulent boundary layers in external flow fields. 

A typical study consists of a turbulent boundary layer on the conical tail of a 

body of revolution. Pate1 et a1.[5] observed that the turbulent boundary layer 

thickens very rapidly and is accompanied by (a) significant variation in static 

pressure across the boundary layer, and (b) a dramatic decrease in the Reynolds 

stress. They concluded that 

(i) the static pressure variation across the layer implies an 

interaction between the turbulent rotational flow in the 

boundary layer and the potential flow outside and that 

prediction of these flows cannot be separated but must be 

done simu~taneous~y, and 

(ii) in order to calculate the development of a thick 

turbulent boundary layer it is necessary to include not only 

the direct effects of pressure but also the indirect effect of 

transverse Curvature on the turbulence as reflected in the 

decrease in mixing length and eddy-viscosi ty. 

Cebeci(61 developed an eddy viscosity formulation for the thick turbulent 

boundary layer. The model was based on experimental measurements and an 

empirical "law of the wall' for flows over thin cylinders and is appropriate to 

external flows only. 

~ ~~~ 
OTY Nozzle Design 

Thick Turbulent Boundary Layer 

Page 3 

While the literature is replete with studies of transverse curvature effects in 

external flows it is equally sparse in studies of internal flows. Patel[”] 

considered an integral method for the calculation of thick axisymmetric 

boundary layers and included an internal flow case. (The flow in a conical 

diffuser was considered; transverse curvature effects were isolated as there 

was no longitudinai curvature.) Patel’s method (an extension of the von Kannan 

integral method for flows with lateral pressure gradients) worked well for 

external flows but was unsuccessful at predicting the internal flow boundary 

layer development. He attributed this to the influence of turbulent terms 

which were omitted from the momentum integral equation. Physically, just as 

the Reynolds stress had diminished in the external flow, Patel expected a 

comespondina increase in the turbulence level due to transverse curvature in 

internal flows. 

The effects of transverse curvature have been discussed frequently in the 

literature but there isn’t much discussion of exactly under what conditions the 

phenomenon becomes important. Generally speaking, most authors state that 

when the thickness of the boundary-layer on a body of revolution becomes of 

the same order as the local radius of the body, the influence of transverse (or 

a 

. lateral) curvature becomes appreciable. 

Some specific data have been reported in the literature, however. Patel et 

al.i51 observed in their experiments at the aft end of a body of revolution that 

0 static pressure remains substantially constant through the 

boundary-layer right up to x/L = 0.90. For x/L > 0.90 the static 

pressure decreases from the wall towards the edge of the 

boundary-] ayer. 

up to x/L = 0.90 the normal component of the velocity is small 

compared with the longitudinal component as required by thin 

boundary-layer theory. By x/L z 0.99, however, the normal component 

of velocity is almost 32% of the longitudinal component at the edge of 

the boundary-1 ayer. 

the streamlines are convex and nearly parallel to the surface inthe 

region x/L < 0.90 where the boundary-layer is thin, and concave and 

divergent over the last 10% of the body length. 
OTY Nozzle Design 

Thick Turbulent Bounday Layer 

Page 4 

Fortunately, Patel et a1.[51 measured the boundary-layer thickness over the 

length of the body. They found that the change from thin to thick 

boundary-layer behavior takes place in the region of x/L = 0.90 where 8/ro is 

approximately 0.62 ( ro is the body radius). 

In another paper, Pateli71 reports on other experiments (not his own) where 

the boundary-layer thickness Is thin compared with the body radius (up to x/L 

= 0.70 (where 8/ro < 0.1 15) and beyond x/L > 0.7 the boundary-layer cannot be 

regarded as thin. These data are quite different from the ones reported above 

and so the results are inconclusive. The difference may be attributable to 

different body shapes and consequently different free-stream velocity 

distribution along the bodies. 

The above-mentioned experiments were co ducted with incompressible fluids. 

We haven't been able to identify any co&able data for experiments with 

compressible fluids. However, White et a1.[8] reported on experiments in 

supersonic flows over slender cylinders and cones. Their results show that the 

skin friction coefficient is very sensitive to the transverse Reynolds number 

(R, = Uwro/ %; ro = body radius) for flow over a Slender cylinder (C, increases 

by up to 100% for Ra = 100 over flat plate values [ R, = MI). Slender cones in 

supersonic flow were similarly shown to have higher skin friction than would 

be predicted by flat plate (thin boundary-layer) theory. 

. 

The above discussion has been limited to boundary-layers in external flows, 

which have received much more attention the literature than internal flows 

with thick boundary-layers. Nevertheless, Whitfield and Lewis(91 reported 

results from experiments on laminar boundary-layer development in nozzles. In 

their experiments the boundary-layer thickness was as high as 90% of the 

nozzle radius. The effects of transverse curvature on the boundary-layer 

thickness were reported to be as large as 15%. These effects should be 

accounted for in the BlM program [3] which already includes the effects Of 

transverse curvature in the mean flow. 
OTV 'Nozzle Design 

Thick Turbulent Boundary Layer 

Page 4 

3. mgitudinal Curvature Effects 

Longitudinal curvature can affect boundary-layer structure through the 

influence of centrifugal force and/or normal stresses when the streamlines are 

curved. Cebeci b Bradshaw[41 discuss the mean flow terms and their order 

of magnitude relative to the other boundary-layer terms. They point out that 

centrifugal force terms are generally intractable and normal stresses are 

usually negligible in weakly turbulent flows. For this reason the thin 

boundary-layer approach is usually adopted even in cases where it isn't 

strictly applicable. Moreover, they claim that attempts to approximate, rather 

than neglect, dp/dy changes the equation type back to elliptic, and methods of 

solution intended for parabolic equations may not work. They do not elabc@&- 

on this nor do they discuss boundary conditions for these flows. 

Normal stresses have received relatively little attention in the literature. 

Finley1 101 analyzed experimental data from cooled-wall and adiabatic 

hypersonic flow nozzles. He attributed the large variation of static pressure in 

hypersonic nozzle boundary-layer flows to be the combined effects of 

longf tudinal curvature of the mean streamlines and the increasing importance, 

as the Mach number rises, of the Reynolds stress contribution to the total 

normal stress perpendicular to the wall. He concluded that at its peak value 

the Reynolds stress may provide a normal stress contribution equal to that of 

the mean static pressure. 

The influence of longitudinal curvature on the mean boundary-layer flow can be 

modeled in several ways. The first way, pointed out by many authors, is to 

simply include the centrifugal force term in the y-momentum equation, to wit 

where R, is the longitudinal radius of curvature of the body. Cebeci and 

Bradshaw[4] point out that the usual assumption (that dp/dy is of the order of 

OIL times the order of i3p/dx) is a because b/L Is small but that the surface 

is flat. The above equation is derived on the premise that flat-surface terms 

are small compared to the 'centrifugal term" which is justified if L/ R, >> b/L. 0 
BRIG:FI!AL PAGE 1s 

OF POOR QUALITY 

OTV Nozzle Design 

Thick Turbulent Boundary Layer 

Page 5 0 

Although this was derived from an order of magnitude analysis it isn't very 

satisfying becasue the probelm of longitudinal curvature is not really 

addressed, that is, its influence may go deeper than just the one term. Indeed, 

that Is what more thorough analyses have shown. 

Two rlgorous approaches to modeling the effects of longitudinal curvature (On 

the mean flow) have been taken in the literature. These are the singular 

perturbation analysis of Van Dyke[ 1 1, 121 and the metric influence of 

curvature method of Schul tz-Srunow and Breued 131. 

The singular perturbation theory is too complex to go into here. Suffice it to 

say that starting with the Navier-Stokes equations the problem is solved by a 

scheme of successive approximations by the method of inner and outer 

expansions. Two complementary expansions are constructed simultaneously, 

and matched in their overlap region of common validity. To lowest order in the 

inner region the Prandtl boundary-layer theory is recovered while the outer 

region is an inviscid flow. Higher-order solutions in each region give the 

perturbations to the boundary-layer and external flows. For example, the 

bounbay-layer displacement effect shows up In both the inner and outer 

regions. The perturbation method simpifies the problem because viscous 

effects are confined to the inner region which is parabolic and hence easier to 

solve than the full (el 1 iptic) Navier-Stokes equations. 

An important point about the perturbation methods is that higher-order 

equations are linear, even though the lowest-order equations may be non-linear. 

This is readily seen in the first- and second-order inner region 

(boundary-layer) equations for incompressible, laminar flow. In terms of the 

stream function and body coordinates (x,y) (see Van Dyke(1 11 for details) 
OTV Nozzle Design 

Thick Turbulent Boundary Layer 

Page 6 

ORIGINAL PAGE 59 

OF POOR QUALITY . 

where the right-hand-side of the first-order equation is the longitudinal 

pressure gradient terms arising from the first-order outer solution. The 

‘inhomogeneous terms’ in the second-order equation include the effects of 

longitudinal curvature, transverse curvature, displacement of the outer flow, 

and external vorticity. The important point is that all these inhomogeneous 

terms depend upon the known first-order solutions. 

The boundary conditions (in addition to the free stream condition 8nd no-slip 

condition) 8re all formally derived in the matching procedure and pose no 

probelms. The method of matched asymptotic expansions is limited only by 

what can be classified as a perturbstion and the number of terms required for 

asymptotic convergence. (Some asymptotic series are actually divergent, a 

point is reached beyond which additional terms increase the error. 

Nevertheless, this does not compromise the validity of the expansion, it merely 

limits it to smaller values of the expansion parameter.) 

The approach of Schul tz-6runow and Breuer[ 131 is considerably different 

from the perturbation approach. They adopt a curvilinear system of coordinates 

and take the arc length along the surface as the x-coordinate. This brings the 

longitudinal curvature directly into the problem (since any differntiation with 

respect to x carries a factor of the ratio of curvature radii fwall position to 

actual position]). Schul tz-Grunow and Breuer derive the boundary-layer 

equations from an order of magnitude analysis performed on the complete 

Navier-Stokes equations. They obtain the following set of equations: 

Conti nui ty: 

where k = lt’Ro. These equations are valid for concave as well 8s convex 0 surfaces (y > 0 for convex walls 8nd y ( 0 for concave walls). 
OTY Nozzle Design 

Thick Turbulent Boundary Layer 

Page 7 

The authors point out that these equations contain all the second-order terms 

of the second-order perturbation theory (Van Dyke[ 1 111, namely those of order 

IIJRe, and the exact metric influence of curvature. An additional term was 

included in the equations as well (the only justification given is that the main 

flow is then an analytic solution of the boundary-layer equations [sic]). Also 

note that these equations reduce to the classical boundary-layer equations 

when k = 0. 

Longitudinal curvature has an effect on the structure of turbulence as well. In 

a boundary layer on a convex wall the centrifugal forces exert a Small 

stabilizing effect. In contrast with that, concave walls have a de-stabilizing 

effect due to instabilites known as Taylor-Gbrtler vortices. Physically, the 

fluid element in a curved stream whose angular momentum decreases with 

increasing distance from the center of curvature is unstable. If the element is 

slightly displaced outward from the center, conserving its angular momentum 

about the center, it will be moving faster than its surroundfngs (which have a 

smaller angular momentum). Therefore the radial pressure gradient that 

controls streamline curvature will be too small to direct the displaced element 

along a streamline of the main flow, and the element will move even further 

outward. The converse argument holds for inward displacement. These 

instabilities affect both the transition to turbulence and the eddy viscosity. 

Bradshaw[ 141 developed an expression for the effect of longitudinal curvature 

on eddy-viscosity based on an analogy between streamline curvature and 

buoyancy in turbulent shear flows (more on this later). 

Cebeci, Hirsch, & Whitelaw[l51 analyzed the turbulent boundary layer on a 

(convex) longltudinally curved surface using the mean flow equations of 

Schul tz-Grunow and Breued 131 discussed above. In their treatment they 

replaced the laminar viscosity with an effective viscosity, Y+<. The eddy 

viscosity (6) was specified to be the standard Cebeci and Smith[ 161 

eddy-viscosity fon?IulatiOn modified by Bradshaw's[ 141 correction for 

longitudinal curvature. 
' OTV Nozzle Design 

Thick Turbulent Boundary Layer 

QRiGlNfal FAG2 ~9 Page 8 

OF POOR QUALITY 

Cebeci and Smith[ 161 point out that streamline curvature may increase or 

decrease the turbulent mixing, depending on the degree of wall curvature, and it 

can strongly affect the skin friction and heat transfer coefficients. They also 

indicate how the streamwise curvature can be incorporsted into the 

eddy-viscosity exrpession by multiplying the right-hand-side of the inner 

eddy-viscosi ty expression by S2 (6radshaw's[ 151 correction), where 

where Ri is analogous to the Rlcherdson number and % is the longitudinal 

radius of curvature. (See the next section for additional discussion of the 

Rlchardson number.) The parameter] is equal to 7 for convex surfaces and 4 

for concave surf8ces, according to meteorological data and by Bradshaw'sf 141 

analogy between streamline curvature and buoyancy. Accordig to Bradshaw, the 

effects of curvature on the mixing length or eddy-viscosity are appreciable if 

the ratio of boundary-layer thickness to radius of curvature, 6 : R, exceeds 

roughly 1 : 300. 

We will complete this discussion of Cebeci et a1.[151 by citing the boundary 

conditions used in their paper: 

where the condition at y = o was obtained assuming v = 0 and vanishing 

vorticity in the free stream and u,(x) is the inviscid velocity distribution. 
OTV' Nozzle Design 

Thick Turbulent Boundary Layer 

page 9 @ 

4. Fluid Stratification Effects 

Fluid stratification (i.e., density gradients) can be of importance in the 

presence of longitudinal curvature. However, the manifestation of density 

gradient effects is sufficiently different that it is discussed separately here. 

The influence of fluid stratification is limited to the eddy-viscosity inasmuch 

as the centrifugal force terms are explicitly included in the mean flow 

equations. In compressible, turbulent flows there is a coupling of the density 

and velocity fluctuations which leads to Reynolds stress-type terms. 

Physically, the turbulence is affected by the interaction of the density gradient 

and centrifugal forces. This interaction can enhance or diminish the turbulent 

mixing depending on the sense of the density gradient. In the extreme case, a 

turbulent flow can be laminar4zed by centrifugal forces. That is, in a stably 

stratified fluid turbulent mixing is suppressed by the restorlng force of the 

pseudo-buoyancy. Conversely, if the fluid is unstably stratified, the turbulent 

mixing is enhanced by the centrifugal force. 

Fluid stratification effects are usually discussed in meteorological 

applications where it is more aptly described in terms of buoyancy forces 

(rather than centrifugal forces). The dimensionless variable usually used to 

describe the influence of buoyancy is the gradient Richardson number (see, for 

example, Turned 1711 

which is the ratio of buoyancy force to shear stress force; g is the 

gravitational acceleration and z is the altitude. This dimensionless Parsmeter 

is readily modified for flows with centrifugal forces by substituting the 

centrifugal acceleration for the gravitational acceleration. Thus we might say 

0 

0 
OTY Nozzle Design 

Thick Turbulent Boundary Layer 

Page 10 

is a modified Richardson number for boundary-layer flow with longitudinal 

curvature. Physically, the centrifugal force would suppress the turbulence on a 

cold, concave surface. 

Observations of fluid stratifiaction effects have been made mostly in 

combustion devices where there are (8) large density gradients and (b) strong 

swirling flows (with centrif.uga1 accelerations exceeding 25,000 9's). 

Accordingly, most of the analyses we are familiar with have been numerical 

solutions of elliptic flows. We have not found any reference to fluid 

stratification effects on longitudinally curved boundary layers. However, 

Bradshaw[l41 does point out that curvature effects on skin friction and heat 

transfer can be much larger at high supersonic speeds. 
OTV Nozzle Design 

Thick Turbulent Boundary Layer 

Page 11 . I) 

5. Recommendations 

OTV nozzles exhibit all the effects discussed above, though it is hard to say 

which of them might dominate. We haven't found anything in the literature to 

account for the increase in turbulent mixing due to transverse curvature in 

internal flows. We have no recommendation to make here except perhaps to 

pursue further literature searching. Likewise, we have found no discussion of 

fluid stratification effects. We recommend calculation of the modified 

Richardson number given in the Previous section and comparison with 

Bradshaw's Rlchardson-like number given in Section 3. This may shed some 

light on the relative importance of longitudinal curvature and density gradient 

ef fects. 

Host likely the strongest influence on the boundary layer comes from the 

longitudinal curvature effect on the lateral pressure distribution in the 

boundary layer. Fortunatley, these effects have been treated adequately in the 

literature. We recommend application of Schultz-Grunow and Breuer'd 131 

formulation for the mean flow equations In conjunction with the Cebeci and 

Smi th(3, 101 eddy-viscosi ty model and Bradshaw'd 141 correction for 

longitudinal curvature. These models are appropriate to both convex and 

concave surfaces. Furthermore, they have been successfully applied to 

turbulent boundary layer calculations in (convex) longitudinally curved surfaces 

by Cebeci et a1.[15]. Finally, these formulations should be compatible with 

the current BLM program (Cebeci[31) in use at Software & Engineering 

Associates, Inc. and can probably be incorporated into the program withmly a 

moderate effort. 

. 
1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

OTV Nozzle Design 

Thick Turbulent Eoundary Layer 

Page 12 

6. References 

AlAA J., Vol. 15, No. 6, pp. 878-881, June 1977. 

BLIMP-J User's Manual, prepared by M. Evans, Aerotherm Division/Acurex 

Corporation, July 1975 under Contract NAS6-30930. 

Weingold, H. D., "The ICRPG Turbulent Boundary Layer (TBL) Reference 

Program,' prepared for ICRPG Performance Standards Working Group, 

July 1968. 

Cebeci, T., 'Boundary Layer Analysis Module,' prepared for Software & 

Engineering Associates, Inc., January 1982. 

Cebeci, T. and Bradshaw, P., Momentum Transfer in Boundary&-, 

McGraw-Hill, New York, 1977. 

Patel, V. C., Nakayama, A., and Damirtn, R., 'Measurements in the 

thick axisymmetric turbulent boundary layer near the tail of a body of 

revolution,'J. Fluid flech., Vol. 63, Part 2, pp.345-367, 1974. 

Cebecl, T., 'Eddy-Viscosity Distribution in Thick Axisymmetric Turbulent 

Boundary Layers,' J. Fluids Engrg., Tt-at-1S8CtiOnS of the ASME, Vol. 95, 

Ser. I, No. 2, pp. 319-326, June 1973. 

Patel, V. C., 'A Simple integral Method fcr the Calculation of Thick 

Axisymmetric Turbulent Boundary Layers,' Aeronautical Quarterly, 

Vol. 25, Part 1, pp. 47-58, February 1974. 

White, F.M., Lessman, R.C., and Christoph, 6.H.. 'Analysis of 

Turbulent Skin Friction in Thick Axisymmetric Boundary Layers,' AI AA J., 

Vol. 1 1, No. 6, pp. 82 1-825, June 1973. 

Whitfield, D.L. and Lewis, C.H., 'Boundary Layer Analysis of 

Low-Densi ty Nozzles, Including Displacement, Slip, and Transverse 

Curvature,' J. Spacecraft and Rockets, Vol. 7, No. 4, pp.462-468, April 

1970. 

Finley, P. J., 'Static Pressure in Hypersonic Nozzle Boundary Layers,' 
OTV Nozzle Design 

Thick Turbulent Boundaf~~ Layer -- 

a Page 13 

1 1. Van Dyke, M,, "Higher approximations in boundary-layer theory, Part 1. 

General analysis," J. Fluid flech., Vol. 14, Part 1, pp. 161- 177, 1962. 

12. Van Dyke, M., Perturbation Methods in Fluid Mec hanic$ Academic Press, 

1964. 

13. Schul tz-6runow, F. and Breuer, W., "Laminar Boundary Layers on 

Cambered Walls," in Basic Oeveloo ments in Fluid Mechanic$ Vol. 1, 

Academic Press, Mew York, 1965. 

14. Bradshaw, P., The analogy between streamline curvature and buoyancy in 

turbulent shear flow," J. Fluid Mech., Vol. 36, Part 1, pp. 177- 19 1, 1969. 

15. Cebeci, T., Hirsch, R. S., and Whitelaw, J. H., "On the Calculation of 

Laminar and Turbulent 6oundary Layers on Longitudlnally Curved 

Surfaces," AlAA J., Vol. 17, No. 4, pp. 434-436, April 1979. 

16. Cebecf, T. and Smith, A. M. O., Analus is of Turbu lent Ro U n &Y m 

Academfc Press, New York, 1974. 

17. Turner, J.S., mymy Effects in Fluids, Cambridge University Press, 

1973. 
ORIGINAL PAGE rg 

OF POOR QUALln 

21 May 1984 

MULTIPLE SHOCK WAVE 

CONSTRUCTION WITH TDK 

Method of Approach 

The logic for shock tracing methods becomes highly. 

complex when two or more shocks are to be traced. However, 

shock tracing methods must be used in order to achieve the 

numerical accuracy required for nozzle performance prediction. 

Four work tasks are proposed below that will give the TDK 

program a capability of dealing with more than one shock wave. 

Task 1) - Start the MOC calculation with a LRC construction. ----- 

The effect of this construction is to ignore the shock 

originating from the tangent point on the attachment circle. 

The present method of RRC construction that ignores the first 

shock by terminating intersecting RRC's can lead to erroneous 

results. The LAC option would be useful when one wants to 

ignore the. tangent point shock and evaluate the effect of a 0 

shock downstream of the attachment point. 

The task consists of implementing the construction 

logic and making it compatible with all the existing options. 

Task 2) Make the RRC construction more reliable. 

The present MOC initial line is more suitable for LRC 

than it is for RRC construction. However, if the shock 

originating from the attachment point is to be traced, one has 

to start with a RRC construction. 

This task consists of investigating the stability of 

the RRC calculation when starting from the initial line and 

making the appropriate changes to have a reliable calculation. 

Task 3) Allow discontinuous slope in wall contour. - 

This task will allow a wall contour to be input with a 

discontinuous slope at one point. This option will be very 

useful when an attached shock is to be analyzed. 
ORIGINAL PAGE 19 

OF POOR QUALITV 

Task 4) Trace two right-running shocks (RRS). ------- 

This task consists of tracing two RRS's where the first 

RRS is induced and the second RRS is an attached shock. The 

intersection of the two RRS's is allowed and may be traced to 

the axis, but the reflected shock will not be treated. Details 

of the intersection of two shocks (RRS - RRS) and (RRS - LRS), 

and of the MOC construction of tracing two RRS, are attached. 
a 

ORIGINAL PAGE fS 

OF POOR QUALITY 
p3c = p3c" a 

a 
. 

14 

i-