A mathematically simple, turbulence closure model designed to treat transonic airfoil flows even with massive separation is described. Numerical solutions of the Reynolds-averaged, Navier-Stokes equations obtained with this closure model are shown to agree well with experiments over a broad range of test conditions

Johnson, D. A.

King, L. S.

English

NASA Technical Reports Server

NASA Technical Memorandum 86826 
NASA-TM-86826 19860004207 
Transonic Separated Flow 
Predictions Based on a 
Mathematically Simple, 
Nonequilibrium Turbulence 
Closure Model 
D.A. Johnson and L.S. King 
October 1985 
NI\S/\ 
National Aeronautics and 
Space Administration 
I IUIII\I \IU IIII UIII UIII UII\ \111\ 1\11 lUI 
NF00069 
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https://ntrs.nasa.gov/search.jsp?R=19860004207 2020-03-20T15:52:53+00:00Z
3 117600187 9825 
NASA Technical Memorandum 86826 
Transonic Separated Flow 
Predictions Based on a 
Mathematically Simple, 
Nonequilibrium Turbulence 
Closure Model 
o A Johnson and L. S King, Ames Research Center, Moffett Field, California 
October 1985 
NI\S/\ 
National Aeronautics and 
Space Administration 
Ames Research Center 
Moffett Field, California 94035 
Transon1c Separated Flow Pred1ct1ons Based on a Mathemat1cally 
SImple, Nonequll1brlum Turbulence Closure Model 
D. A. JOHNSON AND L. S. KING 
NASA Ames Research Center 
Moffett FIeld, CA 94035 
Summary 
A mathemat1cally sImple, turbulence closure model deslgned to treat tran-
sonlc'alrfoll flows even wIth maSSIve separatIon IS descrIbed. NumerIcal 
SolutIons of the Reynolds-averaged, Navler-Stokes equatIons obtaIned w~th 
th1S closure model are shown to agree well wIth experlments over a broad 
range of test condltlons. 
IntroductIon 
The greatest Ilmlt1ng factor in the accurate numerIcal predIctIon of aIr-
fOIl flows has been the lack of an adequate turbulence closure model. For 
alrfo11 flows wIth sufflc1ently mIld pressure gradIents and no separatlon, 
sImple algebraIc turbulence models have been shown to YIeld good 
results. As speeds In the transon1C regime and angles of attack are 
1ncreased, however, adverse pressure gradIents become stronger and sepa-
rated flow regIons make the1r appearance. Under these harsher condItIons, 
the algebra1c turbulence models do not do well. Such cases also present a 
severe test for more SOphlstlcated turbulence models. In Ref. 1 for exam-
ple, relatIvely poor predIctIons of surface pressure were obtalned for a 
serIes of transonIC, InVlscld-V1SCOUS Interaction flows (all w1th some 
separatIon) WIth both the CebecI-Smlth [2], algebralc model and the Jones-
Launder [3], k - £ model. 
Recently (Ref. 4), a mathematlcally SImple closure model was proposed that 
showed promIse 1n treatIng the turbulent boundary layer development even 
under condItIons where the lnvlscld-VlSCOUS InteractIon results In mass1ve 
separat1on. Subsequent to thIS work, further evaluatIons of thIS closure 
model (Refs. 5 and 6) have been made USIng tIme-marchIng, Navler-Stokes 
methods. 
ThIS paper fIrst descrIbes the turbulence closure model of Ref. 4, and 
presents some of the results obtained to date WIth thIS closure model. 
1 
Closure Model 
The intent of the study (described in Ref. 4) was to develop a turbulence 
closure model for a particular class of flows: wall-bounded shear flows 
subjected to large and rapidly changing streamwise pressure grad1ents. 
The impetus was the prediction of turbulent boundary layer development on 
a1rfoils under conditions of strong inviscid-viscous 1nteraction. The 
model was developed under the assumption that the major difficulty encoun-
tered in the prediction of wall-bounded shear flows subjected to adverse 
pressure gradients suff1ciently strong to cause separat10n has been due to 
the lack of (or an improper accounting of) "hIstory effects" OWing to 
convection. 
The closure model can best be described as a hybr1d eddy-v1SCos1ty/ 
Reynolds-shear-stress model. To account for the strong "history effects" 
present in flows with large and rapidly changing streamw1se pressure gra-
d1ents, a simpl1fied Reynolds-shear-stress equat10n (an ord1nary d1fferen-
tial equat10n for the max1mum Reynolds shear stress) 1S used to determ1ne 
eddy-v1SCos1ty changes 1n the streamW1se d1rect1on. EddY-V1scos1ty models 
tend to predict too rap1d a change in the Reynolds shear stress when the 
mean flow f1eld 1S rap1dly d1storted. The present closure model was 
designed to overcome this weakness. 
An algebraic eddy ViSCOS1ty distr1bution 
\It = "to(l - exp - "ti/"to) ( 1 ) 
\lti = 0
2 O.4y(-u'v') 1/2 
m 
(2) 
"'to = a(x).O.0168 u 6*/[1 + 5.5(y/6)6] e 1 (3 ) 
1S used to describe the variat10n of the Reynolds shear stress normal to 
the shear layer under both attached and separated flow cond1tions. "H1S-
tory effects" are taken Into account through the parameter a(x) 10 
Eq. (3). The streamwise d1str1bution of this parameter is establ1shed 1n 
the solution procedure (deta1ls are glven 1n Ref. 5) to give a streamw1se 
d1stribution of -u'v' which sat1sfies the follow1ng ordinary d1fferen-
tial equation: 
2 
a 1 [( )1/2 1/2~ 
= L Tm Teql - T m -
m m 
(4) 
In this equation, T is used to represent -u'v', WhlCh, for convenlence, 
WIll be referred to as the Reynolds shear stress. The subscrIpt m and 
vertIcal slash m denote that the quantIty is evaluated at Ym' the y 
locatIon where -u'v' IS a maxImum. This equatIon whIch descrIbes the 
development of the Reynolds shear stress along the path of maximum kInetIc 
energy IS obtaIned from the turbulence kInetIc energy equatIon. In 
Eq. (4), a 1 = -u'v'/k (where k is the turbulence kInetIc energy) and IS m m 
assumed to be a constant. Lm IS the disSIpatIon length scale, WhICh IS 
modeled algebraIcally. ~m IS the turbulent dIffusIon and Teq IS the 
Reynolds shear stress based on the eddy VIscosIty gIven by Eqs. (1)-(3) 
wIth a(x) equal to unIty. ~m IS assumed to always be posItive and to 
depend on the nonequillbrium state of the flow (I.e., the degree a(x) 
devIates from unIty). The modeled form of thIS term and its formulatIon 
can be found In Ref. 5. When the convectIve term In Eq. (4), umdTm/dx 
becomes small, a(x) approaches unIty and Tm approaches T I. 
eq m 
Results 
Results are now presented whIch demonstrate the present closure model's 
abIlIty to descrIbe wall-bounded shear flows over a range of condItions. 
All of the test cases to be presented have some separated flow; most have 
maSSIve separatIon zones. 
The fIrst flow to be consIdered IS the low-speed dIffuser flow of 
Ref. 1. In FIg. 1, mean velocIty profIles predIcted wIth the present 
closure model and the Cebecl-Smlth model are compared wIth the experImen-
tal measurements. These predIctIons were obtaIned wIth an Inverse-
boundary-layer method. Figure 1 illustrates the baSIC Inadequacy of the 
(equIlibrIum) Cebecl-Smlth model In predIctIng separated flows, and the 
Improvements achIevable wIth the present sImple nonequillbrium model. The 
dIfferent characterIstIC shapes of the mean velOCIty profiles predIcted by 
these two closure mOdels are primarily a result of the eddy ViSCOSIty 
expreSSIons used for the Inner part of the boundary layer. The present 
closure model also dld much better than the Cebecl-Smlth model In predIct-
Ing the surface pressure dIstrIbutIon (Ref. 4). 
3 
16 
12 
E 
u 8 
.;:. 
4 
o 
CLOSURE MODEL 
PRESENT ~ CEBECI·SMITH 
0 EXPERIMENT 0»»d 
x = 3.01 m x = 342 m 
.2 .4 .8 1.0 1.2 -.2 0 .2 .4 .6 .8 1.0 
32 
24 
16 
8 
x = 3.97 m 
o .2 .4 .6 .8 1.0 
ii/ue 
Fig.l. Mean-veloc1ty comparisons for the low-speed diffuser of Slmpson 
et al. [1] a) Upstream of separation; b) at separation; c) downstream of 
separation. 
The remaining examples are all for transonic cond1tions. The f1rst of 
these 1S the flow developed around an aXlsymmetric bump (Ref. 8). In 
F1g. 2, experimental surface pressure d1strlbutlons for three freestream 
Mach numbers (Moo = 0.6, 0.875, and 0.925) are compared w1th computed d1S-
tr1but1ons obta1ned with three d1fferent closure models. The exper1mental 
separation and reattachment locat1ons are also shown 1n thls f1gure. The 
computations were obtained with a computer code based on MacCormack's [9] 
expl1c1t-lmplic1t algor1thm. The numer1cal deta1ls can be found 1n 
Refs. 1 and 5. As eV1dent from Fig. 2, the pressure distrlbutlons 
obtained with the present model are 1n much better agreement w1th exper1-
ment than those obtained wlth elther the Cebecl-Smith or Jones-Launder 
model. 
The pred1cted mean veloclty and Reynolds shear stress prof1les for 
Moo = 0.875 based on the present model are compared to exper1ment 1n 
F1gS. 3 and 4. At the statlons x/c = 0.69 to 0.94, the calculated 
results are for statlons sllghtly upstream of the exper1mental stat1ons. 
This Sh1ft was applied to approxlmately account for the pred1cted shock 
location be1ng 0.02 chord forward of the exper1ment. The Reynolds shear 
stresses are compared 1n shear-layer coord1nates. The 1mportance of com-
par1ng 1n shear-layer coord1nates when the flow 1S h1ghly non1sotropic and 
the closure model is capable of descr1b1ng the Reynolds shear stresses 
accurately but not the Reynolds normal stresses is d1scussed 1n Ref. 5. 
4 
75 
50 
-C P 
25 
0 
50 
-C P 
25 
0 
50 
-Cp 
25 
0 
- 25 
. 4 
Moo 
-
.6 .8 
X SHOCK r / SEPARATED 
/FLOW 
CLOSURE MODEL 
PRESENT 
JONES-LAUNDER 
CEBECI-SMITH 
o EXPERIMENT 
S SEPARATION 
R REATTACHMENT 
Moo = 0 6--.....nI""IIlf[J'"'t:T 
10 
x/c 
R 
12 
Moo = 0 875 
Moo = 0.925 
14 16 
Flg.2. Surface pressure comparisons for the aX1symmetric-bump flow of 
Bachalo and Johnson [8]. 
The mean veloclty and Reynolds shear stress proflles pred1cted by the 
Cebecl-Sm1th and Jones-Launder models (not shown here) do not agree nearly 
as well w1th the exper1mental results (Ref. 5). Both of these models 
pred1cted more rapld r1ses 1n the Reynolds shear stresses at the shock 
and, as a result, underpredicted the momentum losses lncurred by the 
boundary layer. 
5 
There is some disagreement between the present calculatlons and the eKper-
iment as evident from Figs. 3 and 4i {1} the predicted Reynolds shear 
stresses are larger than the experimental stresses in the viclnlty of the 
shock {x/c = 0.69 and 0.75}, and (2) the predicted mean veloclty recovery 
is less than the eKperlment downstream of reattachment.--Ihe former disa-
greement may be due to posslble measurement errors in (u,2 - v,2). At 
x/c = 0.69 and 0.75, the untransformed shear stresses were as large or 
sllghtly larger than the computed values. But, Slnce the measured values 
o EXPERIMENT - PRESENT MODEL 
25 
x/c = 0.63 
20 
E 1.5 
u 
>. 1.0 
5 
0 
0 0 0 .5 1.0 
2.5 
x/c = 0 94 
2.0 
E 15 
u 
>- 10 
5 
0 
0 0 0 0 0 .5 
u/uoo 
Flg.3. Mean-veloclty comparisons for the aXlsymmetrlc-bump flow of 
Bachalo and Johnson [8]i Mm = 0.875. 
6 
1.0 
o EXPERIMENT -- PRESENT MODEL 
25 
x/c = 0.63 0.69 
2.0 
15 
E 0 0 CJ 
>. 1.0 0 0 
0 0 
5 0 0 
0 
0 0 0 0 0 5 10 
25 
20 
15 
E 
CJ 
>10 
0 0 
.5 0 0 
0 
0 0 0 0 5 10 
-u'v'/u~ X 103 
Flg.4. Reynolds shear-stress comparIsons for the aXIsymmetric-bump flow 
of Bachalo and Johnson [B]; Mm = 0.B75. 
of (u ,2 - v,2) were so 1 th d t t t f 1 arge In IS regIon, a coor Ina e ro a Ion 0 on y 
a few degrees resulted In the much smaller shear stresses shown In 
FIg. 4. The latter dIsagreement downstream of reattachment appears to be 
a result of the actual Inner length scales beIng larger than was mod-
eled. The experImental eddy VISCOSIty profIles suggest that the length 
scales In thIS regIon were closer to O.By than the 0.4y assumed in 
Eq. (2). (These discrepancIes between the computatIons and the experIment 
are dIscussed In more detail in Ref. 5.) 
7 
The next two examples are transonic airfoil flows. Solutions for these 
flows were obtained uSIng a Navler-Stokes aIrfoil code based on the algo-
rIthm of Beam and Warming [10]. The two examples are a supercrltical 
airfoIl (DSMA 671) sectIon at transonIC crUIse condlt1ons, and a NACA 
64A010 aIrfoIl section under shock-Induced stall condItIons. The experI-
mental results for these two alrfo1ls are gIven In Refs. 11 and 12, 
respectIvely. The calculatIons were performed at the set angles of attack 
with tunnel wall effects taken 1nto account by uSIng statIc pressure mea-
surements at one chord above and below the aIrfOIls as boundary condI-
tIons. The numerIcal detaIls are given 1n Ref. 6. 
In FIg. 5, predIcted surface pressure dIstrIbutIons obtaIned WIth the 
BaldWIn-Lomax [13] model and the present model for the supercrltical aIr-
fOIl case are compared WIth the experIment. The s~ock wave was weak and 
had lIttle effect on the boundary layer, but the lnVlscld-VlSCOUS Interac-
tIon near the aIrfOIl traIlIng edge was qUIte strong. For example, the 
boundary-layer shape factor H grew to a value of 4 on the aIrfOIl upper 
175 
1.50 
125 
100 
.75 
50 
-Cp .25 
o 
-.25 
-.50 
-75 
-100 
CLOSURE MODEL 
PRESENT 
BALDWI N· LOMAX 
o EXPERIMENT 
T E DETAIL 
-2 
1.0 
-1.25 L---L_--L._....L...-_.l.---1_--'-_....L...-_.L..--.l_--' 
o .1 .2 .3 .4 .5 .6 
x/c 
7 8 9 1.0 
Flg.5. Surface pressure comparIsons at Moo = 0.72 and a = 4.3 0 for the 
DSHA 671 supercrltlcal aIrfoil sect10n (data from Johnson and SpaId [11]). 
8 
surface (separation occurred at x/c = 0.98). The present model was 
better able to predlct the observed rapld thickenlng of the boundary layer 
near the tralling edge than the Baldwln-Lomax model. As a consequence, 
the present model as seen In Flg. 5 gave better predlct10ns of surface 
pressure on the rearward part of the alrfoll. Notlce that the present 
model was able to predict the pressure plateau just upstream of the 
traliing edge caused by boundary layer separatIon. 
The last example IS the NACA 64A010 at shock-lnduced stall conditlons. In 
Flg. 6, predlcted surface pressures are compared wlth the experlment. For 
thiS test case, a CebecI-Sm1th model Solutlon is shown Slnce a steady 
Solutlon could not be obtained wlth the Baldwin-Lomax model. As eVldent 
1.50 
125 
1.00 
.75 
.50 
-Cp 25 
0 
- 25 
-.50 
-.75 
-1.00 
-125 
0 
c: 
CLOSURE MODEL 
-- PRESENT 
--- CEBECI·SMITH 
o EXPERIMENT 
.1 .2 .3 .4 .5 .6 .7 .8 
x/c 
.9 1.0 
Fig.6. Surface pressure comparlsons at Moo = 0.8 and a = 6.20 for the 
NACA 64A010 airfoll section (data from Johnson and Bachalo [12]). 
from Flg. 6, the surface pressure distrlbutlon obtalned wIth the present 
model IS In substantlally better agreement wlth the experlment than that 
obtalned wlth the CebecI-Smith model. In F1g. 7, pred1cted mean velocity 
and Reynolds shear stress proflles are compared with the exper1mental data 
1n shear-layer coordlnates. The pred1cted results based on the present 
model compare qUlte favorably wlth the experlment, especlally conslderlng 
the dlfflculty of this test case and the experlmental uncertainties. 
9 
u 
-->-
o EXPERIMENT - PRESENT MODEL 
x/c = 0 67 
.15 
.10 
05 
o L-.l4"F-...I.----.J.--' 
CEBECI·SMITH MODEL 
x/c = 0.83 
-.5 0 .5 1.01.5 -10 0 10 20 30 -.5 0 .5 1.0 15 -10 0 10 20 30 
u/uoo 
F1g.7. Mean-veloc1ty and Reynolds shear stress comparisons at Mm = 0.8 
and a = 6.2 0 for the NACA 64A010 airfoil section (data from Johnson and 
Bachalo [12]). 
Concluding Remarks 
As illustrated by the foregoing examples, a simple closure model has been 
developed which is capable of describ1ng wall-bounded shear flows 
developing on a1rfo1ls at transon1c conditions even when maSSlve 
separatlon 1S present. The closure model requires llttle more 
computational effort than equ1llbrium algebraic closure models, and does 
not introduce any numerical stability problems. 
References 
1. Horstman, C.C.; Johnson, D.A.: Predict10n of Transonlc Separated 
Flow. AIAA Journal, vol. 22, no. 7 (July 1984) 1001-1003. 
2. Cebec1, T.; Smlth, A.M.O.: Analysis for Turbulent Boundary Layers. 
New York, Academ1c Press, 1974. 
3. Jones, W.P.; Launder, H.E.: The Prediction of Laminarizatlon wlth a 
Two-Equatlon Model of Turbulence, Internatlonal Journal of Heat 
Transfer, Vol. 15 (Feb. 1972) 301-314. 
4. Johnson, D.A.; K1ng, L.S.: A New Turbulence Closure Model for 
Boundary Layer Flows wlth Strong Adverse Pressure Gradlents and 
Separat1on, AIAA Paper 84-0175, Reno, Nev. (1984). 
5. Johnson, D.A.: Predictions of Transonic Separated Flow with an Eddy-
V1SCos1ty/Reynolds-Shear-Stress Closure Model, AIAA Paper 85-1683, 
C1ncinnat1, Oh10 (1985). 
6. K1ng, L.S.; Johnson, D.A.: Separated Transonic Airfoil Calculations 
w1th a Nonequilibrium Turbulence Model, NASA TM-86830, 1985. 
10 
7. Simpson, R.L.; Chew, Y.T.; Shivaprasad, B.G.: The Structure of a 
Separating Turbulent Boundary Layer. Part I. Mean Flow and Reynolds 
Stresses, Journal of Fluid MechaniCS, vol. 113 (1981) 23-51. 
8. Bachalo, W.O.; Johnson, O.A.: An Investigation of TransonIC 
Turbulent Boundary Layer Separatlon on an AXlsymmetrlc Flow Model, 
AIAA Paper 79-1479, Wllllamsburg, Va. (1979). 
9. MacCormack, R.W.: A Numerical Method for Solving the Equations of 
Compressible VISCOUS Flow. AIAA Journal, vol. 20 (Sept. 1982) 
1275-1281. 
10. Beam, R.; Warming, R.F.: An Implicit Factored Scheme for the 
Compressible Navler-Stokes Equations, Journal of Computational 
PhYSiCS, vol. 15 (1974) 299-319. 
11. Johnson, O.A.j Spaid, F.W.: Supercrltical Airfoil Boundary-Layer and 
Near-Wake Measurements, Journal of Aircraft, vol. 20 (Apr. 1983) 
298-305. 
12. Johnson, D.A.; Bachalo, W.D.: TransoniC Flow Past a Symmetrical 
Alrfoll--Invlscld and Turbulent Flow Properties, AIAA Journal, 
vol. 18 (Jan. 1980) 16-24. 
13. Baldwin, B.S.; Lomax, H.: Thin Layer Approximation and AlgebraiC 
Model for Separated Turbulent Flows, AIAA Paper 78-257, Huntsville, 
Ala. (1978). 
11 
1 Report No I 2 Government Accession No 3 Recipient's Catalog No NASA TM 86826 
4 Title and Subtitle 5 Report Date 
TRANSONIC SEPARATED flOW PREDICTIONS BASED ON A October lqS'l 
MATHEMATICALLY SIMPLE, NONEQUILIBRIUM TURBULENCE 6 Performing Organization Code 
CLOSURE MODEL 
7 Author(s) 8 Performing Organization Report No 
D. A. Johnson and L. S. King 
10 Work Unot No 
9 Performing Organization Name and Address 85390 
Ames Research Center 11 Contract or Grant No 
Moffett Field, CA 94035 
13 Type of Report and Period Covered 
12 Sponsoring Agency Name and Address Technical Memorandum 
National Aeronautics and Space Administration 14 Sponsorong Agency Code 
Washington, DC 20546 505-51-11 
15 Supplementary Notes 
Point of Contact: D. A. Johnson, Ames Research Center, MS 229-1, Moffett 
Field, CA 94035 (415) 694-5397 or FTS 448-5397 
16 Abstract 
A mathematically simple, turbulence closure model designed to treat 
transonic airfoil flows even with massive separation is described. Numerical 
solutions of the Reynolds-averaged, Navier-Stokes equations obtained w1th 
this closure model are shown to agree well w1th experiments over a broad 
range of test conditions. 
17 Key Words (Suggested by Author(s)) 18 O,strobut,on Statement 
Transonic flow prediction Unlimited Turbulence closure 
Separated flow Subject category - 34 
19 Securoty Casslf (of thiS report) 120 Securoty Classlf (of thiS page) 21 No of Pages 22 Price' 
Unclassified Unclassified 13 A02 
'For sale by the National Technocal Information Service, Springfield, Vlrglnoa 22161 
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Transonic separated flow predictions based on a mathematically simple, nonequilibrium turbulence closure model

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