Table Of ContentPREDICTION OF VERY HIGH REYNOLDS NUMBER COMPRESSIBLE SKIN FRICTION
John R. Carlson"
NASA Langley Research Center,
Hampton, Virginia 2368 1
ABSTRACT local skin friction coefficient, z,/q,
c f
D skin friction drag
Flat plate skin friction calculations over a range of
F shape function
Mach numbers from 0.4 to 3.5 at Reynolds numbers ,
near-wall damping function for K E
from 16 million to 492 million using a Navier Stokes Hf total enthalpy ~
method with advanced turbulence modeling are com-
I freestream turbulence intensity
pared with incompressible skin friction coefficient cor-
K turbulent kinetic energy
relations. The semi-empirical correlation theories of van
k mixing-length constant
Driest; Cope; Winkler and Cha; and Sommer and Short
L length of flat plate, 5-m
T' are used to transform the predicted skin friction coef-
1 mixing length
ficients of solutions using two algebraic Reynolds stress
M Mach number
turbulence models in the Navier-Stokes method
N number of grid points
PAB3D. In general, the predicted skin friction coeffi-
n distance normal to wall
cients scaled well with each reference temperature the- -
P production term for turbulent kinetic
ory though, overall the theory by Sommer and Short
energy
appeared to best collapse the predicted coefficients. At
P static pressure, Pa
the lower Reynolds number 3 to 30 million, both the
4 dynamic pressure, Pa
Girimaji and Shih, Zhu and Lumley turbulence models
Reynolds number, (puL)/p
predicted skin-friction coefficients within 2% of the RL
R,* transformed Reynolds number
semi-empirical correlation skin friction coefficients. At
turbulent Reynolds number, K2/(v&)
the higher Reynolds numbers of 100 to 500 million, the RT
S strain tensor
turbulence models by Shih, Zhu and Lumley and Giri-
sc Sutherland's constant, 110.33 K
maji predicted coefficients that were 6% less and 10%
T temperature
greater, respectively, than the semi-empirical coeffi-
T' intermediate reference temperature
cients.
t time
NOMENCLATURE
U velocity
-
A = surface area, CAi U magnitude of velocity,
A,B = temperature ratio constants for Van uk Cartesian velocity4 co3mp,one nts
Driest equation friction velocity,
u7
Ai = incremental surface area U law-of-the-wall coordinate, U/u,
a = speed of sound W vorticity tensor
CF = average skin friction coefficient X streamwise distance
c; = transformed average skin friction X. Cartesian displacement components
5
coefficient law-of-the-wall coordinate, (nu,)/v
y+
CEI = constants for K-e equations law-of-the-wall height of first cell
Yl
CE27 Cp Z vertical distance
a free parameter for K turbulent tripping
profile
*Senior Scientist, Aero- and Gas-Dynamics Division. 6 boundary 1a yer thickness
E turbulent dissipation
Copyright 1998 by the American Institute of Aeronautics and Astro- Y ratio of specific heats, 1.4
nautics, Inc. No copyright is asserted in the United States under Title
0 boundary layer momentum thickness
17, U.S. Code, the U.S. Government has a royalty-free license to exer-
cise all rights under the copyright claimed herein for Governmental K von Karman constant
purposes. All other rights reserved by the copyright owner. P laminar viscosity
1
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turbulent viscosity parisons were Mach numbers from 0.4 to 3.5 at unit
viscosity evaluated at T' Reynolds numbers of 1 to 30 million per foot. Skin fric-
kinematic viscosity, ,u/p tion predictions are compared with the semi-empirical
density theories. Estimations of solution convergence and errors
density evaluated at T' and ps are discussed. Different transformations of Reynolds
shear stress number and skin friction are used for several compari-
viscosity power law power, Eqn. 24 sons with the skin friction theories.
Subscripts COMPUTATIONAL PROCEDURE
adiabatic wall Governing Equations
cor = correlation reference conditions
The general three-dimensional Navier-Stokes
i = incompressible
method PAB3D version 13 was used. This code has sev-
L = laminar
era1 computational schemes and different turbulence
T = turbulent
and viscous stress model^.^-'^ The governing equations
T' = based on temperature T'
are the RANS equations obtained by neglecting all
t = freestream total conditions
streamwise derivatives of the viscous terms. The result-
W = conditions at the wall surface
ing equations are written in generalized coordinates and
6 = conditions at the boundary layer edge
conservative form. Viscous model options include thin-
=, 0 = freestream conditions
layer assumptions in any direction or any two indices
fully coupled with the third uncoupled. Typically, the
INTRODUCTION
fully three-dimensional viscous stresses are reduced to a
The efficiency of airplane design has improved thin-layer assumption, but this assumption may not
considerably as computing power and computer pro- always be appropriate. Experiments such as the investi-
grams have advanced and specialized tools, such as gation of supersonic flow in a square duct were found to
inverse design methods and advanced graphic inter- require fully coupled two-directional viscosity to prop-
faces, have been developed. Despite this, some funda- erly resolve the physics of the secondary cross-flows.6
mental aerodynamic flow issues continue to elude both The Roe upwind scheme with third-order accuracy
the experimental- and the computational-based is used in evaluating the explicit part of the governing
researcher. One such issue involves several aspects of equations, and the van Leer scheme is used to construct
skin friction, specifically, the measurement of skin fric- the implicit operator. The diffusion terms are centrally
tion experimentally; the determination of skin friction differenced, and the inviscid flux terms are upwind dif-
through parametric correlations; and the prediction of ferenced. Two finite volume flux-splitting schemes are
skin friction using advanced computational methods. A used to construct the convective flux terms. The code
survey of some of the semi-empirical theories of skin can utilize min-mod, van Albeda, Spekreijse-Venkat, or
friction can be found in (Refs. 1-5). Most of the semi- modified Spekreijse-Venkat limiters. All solutions were
empirical theories were fit to data over a limited range developed with the third-order-accurate scheme for the
of Mach number and Reynolds number and have had convective terms and second-order scheme for the vis-
varying degrees of success in obtaining accurate corre- cous diffusion terms. The min-mod limiter was utilized
lations. Typically 5 to 10% error is quoted for the empir- in the blocks containing wall-bounded flow, otherwise
ical determination of skin friction due in part to scatter the van Albeda limiter was used.
and accuracy in the experimental data sets, corrections The code can utilize a 2-, 3-factor, or diagonaliza-
for model effects and test techniques, and to a small tion numerical scheme to solve the flow equations. The
degree, simplifications made in deriving the theories. 2-factor scheme can be used when the predominant flow
The theories chosen for these comparisons will be direction is oriented along the i-index of the grid. An
those of Van Driest, Cope, Winkler and Cha, and Som- example would be a jet-plume or nozzle configuration
mer and Short.' These correlations will be compared where the j k index grids generally represent cross-
~
with results from a three-dimensional Reynolds-aver- planes of the exhaust flow. Though this scheme
aged Navier-Stokes (RANS) code PAB3D (Refs. 6-10) typically requires 10-15% less memory than the
using explicit algebraic Reynolds stress turbulence 3-factor scheme it is less applicable to many general
models for calculations on a 5-meter flat plate with zero 3-D aerodynamics problems due to inconsistency
pressure gradient. The conditions used for these com- between the mesh topologies and the flow solution.
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These flow simulations were performed using the 3-fac- and Lumley is based on the turbulent constitutive rela-
tor scheme. tions developed by Shih and Lumley." The model by
Girimaji is also based on a set of algebraic relations
Turbulence Simulation between the turbulent Reynolds stresses and the mean
velocity field but uses the pressure/strain relationship by
Version 13 of the PAB3D code used in this study
Speziale et al.I3 The model is similar to that of Gatski
has options for several algebraic Reynolds stress (ASM)
and SpezialeI6 except for the determination of the vari-
turbulence simulations. The standard model coefficients able coefficient C, . Further discussion of the turbulence
of the K-E equations were used as the basis for all of the
model equations and the algebraic Reynolds stress tur-
linear and nonlinear turbulent simulations as shown in
bulence model implementation can be found in (Ref. 9).
Table 1.
Turbulent Trip Tactics
Table 1. Linear K - E Standard Coefficients
The tripping of laminar flow to turbulent can be
Constant Value
fixed through the imposition of E and K profiles at
1.44 user-specified points or grid lines. The line or plane of
CEI
the specified trip area is surveyed for the maximum and
1.92
CE2 minimum velocity and vorticity, and a shape function
0.09 from 0 to 1 is created. The shape function, F, is defined
CU
as
The near wall damping function of Launder and
Sharma," F = f ~ fmin
fmax fmin
fu = eXp[-3.41/(1 + R~/50)~] where
determined the behavior of E as a function of f = UlWl
R, = K2/(v&). The boundary conditions for E and K
at the wall are and
and f is the product of the velocity, U , and vorticity magni-
tude, IWI . The turbulent kinetic energy profile is then
K, = 0 generated using
The turbulence model equations are uncoupled from the K = aOF
RANS equations and are solved at the same time step as
that of the mean flow solution. Relatively high Courant- where a is a free parameter that determines the magni-
Friedrichs-Lewy (CFL) numbers can be used (e.g. tude of E and K profiles as a percent of the local veloc-
1 < CFL < 10 ) and though rather problem dependent, ity magnitude 0.T he value used for this paper is 0.1%
occasionally flow solution transients can force a tempo- (a = 0.001 ). The E profile is developed from the
rary time step reduction of the solution of the turbulence assumption that production over dissipation of
equations. More often it is a grid-resolution or grid- turbulence is 1, that is, P/E = 1 . This results in the
quality issue rather than strictly a turbulence modeling equation
difficulty that requires lower CFL numbers to be used.
The turbulence equations are solved at all grid levels, = 2C,K2Sij(aui/axj) (1)
not just at the finest grid level. The result of the tripping is typically observed as a
The algebraic Reynolds stress turbulence model by localized spike in the K field. Depending upon the flow
GirimajiI2 with the Speziale, Sarkar and Gatski (SSG) conditions, such as local Reynolds number, momentum
coefficient^'^ and the model by Shih, Zhu and LumleyI4 Reynolds number, or freestream Mach number, turbu-
(SZL) were utilized in this study. The coefficients of the lence may or may not develop downstream of the trip
linear K E model were used unmodified as there has point. Turbulence quantities such as the production P ,
~ _ -
not yet been a recalibration performed with any of the or the turbulent stresses u'u',u'v.'.,. are left as floating
ASM's in this code. The model developed by Shih, Zhu point numbers and are not explicitly set to zero or any
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other value “upstream” of trip points. The initial levels Karman-Schoenherr Eauation
of these quantities are determined by thresholds of
A number of semi-empirical correlations have been
&/a and pT/p that are parameters in a user input file.
derived for skin friction coefficients with Reynolds
Table 2 lists the limits used for these calculations.
number. The basis for most of the correlations in this
report is a relationship by Schoenherr derived from the
Table 2. Numerical Thresholds for Turbulence
work of von Karman.18 A numerical fit by von Karman
Parameters
to a set of experimental data resulted in
Constant Value
0.0001 E -- 4.13 loglo[C,R,] (4)
0.1
C, and R, are defined here as
Under some circumstances, these thresholds can be
(5)
manipulated so as to cause a laminar boundary layer to
transition without any explicit tripping specified. As an
example, given the freestream conditions of M = 0.4, and
T, = 551.8R, and a unit Reynolds number of 1 million,
with &/a = 0.0004, the flat plate flow was fully tur-
bulent. If &/a = 0.0001 , transition never occurred. As
a point of interest, the lower limit of K can be related to
though not explicitly defined (Ref. l), total drag is
the freestream turbulence intensity, I , as
defined as the summation of the incremental shear stress
at the wall times the incremental area.
So that &/a = 0.0001 would correspond to a fairly 1
low freestream turbulence intensity of I = 0.02% . A pro-
Equation 4 is a result of a number of simplifications
posed lower limit of freestream turbulence intensity to
and assumptions about the character of an incompress-
significantly influence transition is 0.08%.17
ible boundary layer. Typically, the viscous sublayer
Unfortunately the use of the particular ratios,
&/a and pT/p, for setting the lower threshold values below y+ = 11.5 , is neglected and the velocity gradient
for the various turbulent quantities makes a definitive at the low end of the log-layer region, 11.5y+ I n I 0.26,
correlation with I difficult because of the relationship is set equal to 0.21 8.19 Additionally, the turbulent stress
in the log-layer region is assumed constant and equal to
between E at its lower threshold and &/a. That is,
the laminar stress at the wall. This assumption simplifies
Cpa2(&/a)2 the solution of the velocity distribution (to be discussed
‘lthreshold (pT/p)p in the following paragraphs) for a specific wall shear-
stress which would be associated with a particular skin-
P T ~wPo uld have to be varied as the square of &/a to friction coefficient and set of free-stream conditions and
maintain a fixed ~l~~~~~~~~~ so potentially a correlation is fairly consistent with the nature of the total stress in
between I and transition to turbulence for this configura- the boundary layer. Figure 1 shows the interplay
tion could then be developed. between the laminar stress and turbulent stress in the
Calculations performed for this paper explicitly set boundary layer for M = 0.4, R = 1 millionift. near the
the trip point at the leading edge of the plate. Depending trailing edge of the flat plate (R, = 15.8~1)0. ~
up the freestream Mach number and the unit Reynolds
The cross-over in the stresses occurs at approxi-
number, transition to turbulence occurred at different
locations downstream of the leading edge. A section in mately y+ = 10 which is fairly close to the previously
stated assumption of 11.5. The log-layer extends to
Results and Discussion will address this issue further.
approximately 0.26 which is in this is case is
y+ = 1400 , and the turbulent stress in the log-layer is
SEMI-EMPIRICAL THEORIES
observed to be slightly less than that of the laminar
The following is a short review of the theories used stress at the wall. The boundary layer velocity profile is
in this paper. Discussions of additional theories can be plotted for visual reference.
found in (Ref. 1).
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where
T B B 1
1
4 = =
A
25 1
20 0.8
‘g 15 0.6 c51 L and
10 0.4 -B = T--aIw
TW
5 0.2 cF,wan d RL,w are defined as:
n” n
10.’ 10’ io’ 10’ io3 io4
Y+
Figure 1. Laminar and turbulent stresses in boundary
layer. and
Historically, either Prandtl’s (I = Kn) or von
Karman’s (I = K(du/dn)/(d2u/dn2) ), theory have been
used for the mixing-length 1 and will produce similar
An alternate form of the factors A and B are published in
forms for the skin friction equation (Eqn. 4) but differ-
(Ref. 19) using Mach number and temperature. Equa-
ent coefficients on either side of the relation. Subse- tions (11) and (12) are equivalent to the following
quent to this, various techniques have been applied to equations, Eqs. (15 ) and (16 ), if the boundary layer edge
deriving the skin-friction relationships. The von Karman conditions, 6, are taken to be the same at the free-
momentum integral (Eqn. 8) is the basis for relating the stream conditions and Taw = T, .
skin friction and Reynolds number. Since the integral
equation is quite intractable analytically, several alter-
nate representations (typically numerical approxima-
tions) of the integral have been derived.”
and
+
1
B = 2
-1 (16)
TWIT_
Van Driest Method
Both Van Driest and the theory by Cope evaluate quanti-
Van Driest’s analysis used Prandtl’s mixing length ties by the conditions at the wall and an outside
and an interpolation expression representing von reference. Reference 1 is not explicit in the outside ref-
Karman’s integral equation considering for the effects erence condition definition. The subscript 6 was defined
of compressibility. This resulted in the following equa- as “conditions outside the boundary layer,” whether that
tion from (Ref.1). implies free-stream conditions or the conditions at the
0.242 @ = logl0[(:F,wRL,w(<) T 1/2 ] (9) pflooiwnt qouf aUnt/iUti_es =0 .50%.99 5le ,s ws hthicahn bfyre dee-sfitnreitaimon, pisla cneo ts odmise-
cernible. For the flat plate cases considered in this
report, the reference shift results in slight shifts in both
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American Institute of Aeronautics and Astronautics
Reynolds number and skin friction, such that the result- conditions as follows:
ant numbers remain generally along the same curve.
Cope Method
The theory by Cope, rather than working from a
and
mixing length law, assumes that the compressible veloc-
ity profile can be transformed to the incompressible pro-
file by using the wall density and viscosity. The
resulting equation is shown.
Each theory has a different set of normalizations
0.242 hglO['.F,wRL,w~] and are shown in Table 3.
~ (17)
W
Table 3. Summary of Theories
Winkler and Cha Method
Theory C; R*L
The theory by Winkler and Cha uses a different
scaling assumption for arriving at the compressible skin Van Driest 'F,w
friction, i.e.,
Cope 'F,w RL, w
TWIT,
and results in the following equation Winkler (~~/~,)1/2 (Tw/Taw)1/4
and Cha
CF( Tw/Taw)1/4 RL (Tt/TF)1/2
Sommer and CF,T, RL,T,
Short T'
where now C, and RL are defined as previously dis-
cussed under the Karman-Schoenherr equation sec-
The skin friction coefficient calculated by the
tion. For this correlation all quantities are evaluated at
Navier-Stokes (NS) method, are non-dimensionalised
free-stream conditions, with the exception of the total
by different coefficients than most of the correlations.
drag integration which the author assumes still utilizes
The definitions for skin friction, C, and Reynolds num-
the viscosity at the wall for the determination of the wall
ber, RL in the computational method are the same as
shear stress.
Eqs. (5)a nd (6); therefore, the transformations shown in
Table 4 are required to compare the computational
Sommer and Short T' method
results with most of the correlations.
The T' method utilizes an empirical relationship
between the Mach number and temperature at the edge Table 4. Summary of computational transformations
of the boundary layer, and the wall temperature to arrive
Correlation
at a reference temperature at which the properties (den-
Reference ',,cor RL, cor
sity and viscosity) of the boundary layer are evaluated.
Several equations have been proposed, as discussed in
(Ref. l), but only the method of Sommer and Short will
be shown.5T he reference temperature is calculated from
T'
(2
1
1 + 0.035 M : + 0.45 - 1
~
Free Stream
The skin friction formula then has the form
0.242
Jcrs ~ log IO [CF,T'RL,T'] (21) For simplicity in the analysis, several theories use
the single power relation of
~
where CFS, and RLS, are now evaluated at the reference
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where typically o = 0.76. A slightly more accurate RESULTS AND DISCUSSION
relation is Sutherland’s law.
Determination of Boundary Layer Edge Criteria
An accurate and consistent determination of the
edge of the boundary layer is important for calculation
of momentum thickness Reynolds number, shape
Figure 2 shows the difference in the viscous ratio
factors, and edge conditions. The skin friction correla-
using the power law compared to Sutherland’s law. The
tion, R, vs cf , was used to evaluate the applicability of
temperature T’ is substituted for T, when those
several velocity and enthalpy edge values. Figures 3
reference conditions are used. For temperature ratios
less than 2, the viscosity scaling error would be 5% or through 5 are the two criteria for M = 0.4 and figures 6
less using the power law compared to Sutherland’s law. through 8 are for M = 1.2 at the Reynolds number of
1 milliodft. The skin friction and Reynolds numbers
Since temperature ratio ranged from 1.02 at M = 0.4 to
plotted here are evaluated using free stream values. At
2.41 at M = 3.5, the Sutherland’s law relation was uti-
lized for post processing the skin friction predictions. M = 0.4, the laminar and turbulent coefficients are con-
sistent up to the velocity edge criteria of 0.995. The
enthalpy edge criteria is consistent up to 0.99. The
Sutherland’s Law
equivalence of the two criteria for this condition is
~~~~~~ Omega Power Law
shown in figure 5.
4
E
3.5 aA Uu//Uu,,== 00..9979
0 U/U,= 0.995
1i2 2.5 00.0.0019
0.008
0.007
0.006
1.5
0.005
0.004
TWIT,
u% 0.003
Figure 2. Viscosity ratio from temperature ratio.
0.002
Solution Process
Turbulent flow solutions that use ASM and the two-
equation linear K - E model require 23 words of mem-
ory per grid point. The code speed is dependent on the
turbulence model, viscous model assumptions, and
numerical schemes. All solutions for this study were Figure 3. Velocity edge criteria M = 0.4, R = 1 million/
ft.
performed on Silicon Graphics workstations. The code
was compiled using Fortran 90, double-precision
(64-bit) with 02 level of optimization. The code speed The transonic case, M = 1.2, required much lower
edge criteria, (Figs. 6 and 7). With the exception of the
at the finest grid level was approximately 110 micro-
first cell, the velocity criteria of 0.98 and enthalpy crite-
secondsliterationigrid point running a 3-factor solution
ria of 0.97 give fairly consistent results for the skin fric-
scheme, 1 thin-layer viscous direction and using an
tion correlation. These two criteria are plotted in figure
algebraic Reynolds stress turbulence model. The com-
8 and show similar results. For the higher supersonic
puter memory requirement was approximately
18 megabytes. conditions of Mach = 2.4 and 3.5, a consistent boundary
layer edge could be determined with the enthalpy crite-
Solution residual and total skin friction were used
ria as high as 0.995. The choice of which edge criteria is
to gauge solution convergence. Total skin friction was
used results in different temperature and viscosity val-
solution converged and grid converged.
ues being used in determination of the boundary layer
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edge conditions. Recall that the semi-empirical theories 0 U/U,=0.96
of Cope and Sommer and Short used boundary layer A U/U,=0.97
D U/U,=O.98
a
edge conditions for the scaling of the skin friction and u/u,=o.99
0 U/U,=O.995
Reynolds number rather than the free-stream. No matter
which criteria is applied, the edge conditions will be
slightly different than the free-stream. The variation of 0.007
0.006
the edge velocity is plotted in figures 9 and 10 for the
0.005
two edge criteria at M = 0.4 and 1.2. The two edge crite- 0.004
ria produce very similar edge velocities for the subsonic
case, but the particularly difficult transonic condition of ok 0.003
M = 1.2, determines significantly different edge veloci-
ties depending upon the criteria chosen. 0.002
0 H/H,=0.95
0 H/H,=0.96
A H/H,=0.97
D H/H,=0.98
a
H/H,=o.~~
H/H,=0.995
0.01
0.009 Figure 6. Velocity edge criteria at M = 1.2, R = 1
0.008
milliodft.
0.007
0.006
0.005
0.004 0 H/H,= 0.95
0 H/H,= 0.96
A H/H,= 0.97
Ok 0.003 D H/H,= 0.98
a
H/H,= 0.99
H/H,= 0.995
0.009
0.008
' 0.007
0.006
0.001 I I 1
" 1 0 '0.0005
lo2 io3 10 10
Re 0.004
Figure 4. Enthalpy edge criteria M = 0.4, R = 1 milliod Ok 0.003
ft.
0.002
WH,=O99
UNO=0 995
0.001
102 103 104 10'
0.009 Re
0.008
0.007
0.006 Figure 7. Enthalpy edge criteria at M = 1.2, R= 1
milliodft.
0.004
O" 0.003 ESTIMATION OF LAMINAR-TO-TURBULENT
FLOW RATIO EFFECTS
0.002 I I I
Laminar flow present in a computational flow solu-
tion changes the predicted skin friction from that of an
assumed fully turbulent flow. For a given physical
geometry, obviously the Reynolds number of the
problem is a major factor in determining the degree of
Figure 5. Equivalence of velocity and enthalpy edge, laminar flow that might exist. Secondarily, the existence
M = 0.4, R = 1 milliodft. of laminar flow in the CFD solution is also dependent
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American Institute of Aeronautics and Astronautics
upon whether there is sufficient grid density to actually
0 HIHo=0.97
predict the laminar flow. Assuming an incompressible 0 urn, =0.98
flow with a critical Reynolds number of 500,000, 1
Table 5 is an estimation of the expected laminar run for
0.995
the 5-m flat plate at different unit Reynolds numbers.
0.99
The third column is an estimation of the percentage of
$
the flow that would be laminar. The fifth and sixth col 0.985
umns are the two terms of an expression for total skin
0.98
friction derived in Schlichting18 accounting for the
0.975
initial laminar region. The first term is the regular
expression for fully turbulent flat plate skin friction and 0.971 0 105 10 107 10
the second term is a correction for the laminar segment. R,
c,
= 0"', 074 A ,5x105 < R, < lo7 FMig =u r1e. 21,0 R. N=o 1r mmaillilzioedd fvte. locity at boundary layer edge,
RL
Figure 1 shows1 the variaAble as afunction of crit-
ical Reynolds number. The quoted Reynolds number
WHO=0 97
UN,=O98 range of applicability is less than lo7, so it must be
noted that applying it to the present problem is an
extrapolation of this equation. At Reynolds numbers
0.007
greater than 4 million per ft., the laminar aspect of the
0.006
flat plate flow becomes less than 1/2 of 1 percent of the
0.005
total length of the plate. Additionally, the estimated
0.004
decrease in total skin friction coefficient is less than
O" 0.003 0.00003. Therefore, a leading edge spacing of 0.01 m
should be sufficient to predict the laminar flow at the
0.002 lower Reynolds numbers. It is inadequate to resolve any
laminar flow in the high Reynolds number range, but the
degree of error in total skin friction coefficient is esti-
mated to be less than 0.00003.
Figure 8. Equivalence of velocity and enthalpy edge,
M = 1.2, R = 1 milliodft.
o WHo=0.99
Urno=0.995
1
0.995
0.99
0.985
0.98
0.975
0.97
io4 10 lo6 10 108
R,
Figure 9. Normalized velocity at boundary layer edge,
M = 0.4. R = 1 milliodft. Figure 11. Variation of constant A with critical
Reynolds number.
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Table 5. Estimation of laminar flow contribution to total skin friction coefficient
Reynolds Reynolds Est. Laminar x/L (%) 0-.074
number number -A/R,
run, x (m) L=5 m
(milliodft.) (million) 5KL
1. 16. 0.1524 3 .O 0.00268 -.00010
2. 32. 0.0762 1.5 0.00233 -.00005
4. 65. 0.0381 0.8 0.00203 -.00003
8. 131. 0.0191 0.4 0.00176 -.00001
15. 246. 0.0102 0.2 0.00155 -.00001
-.ooooo
30. 492. 0.005 1 0.1 0.00135
Flat-Plate Grid Table 7. Grid dimensions
Block i-dim j-dim k-dim
The 5-m flat-plate multiblock grid had an H-type
mesh topology, with three blocks placed streamwise. 1 11 2 121
The computational domain included an inflow block, 2 61 2 121
block 1, extending 2.5 m upstream from the leading 3 13 2 121
edge of the 5-m flat plate. The plate, block 2, had an
initial streamwise grid spacing at the leading edge of
0.01 m and was exponentially stretched from the leading Grid Convergence
edge to the trailing edge at a rate of 6.7% using a total of
61 grid points. Block 3, downstream of block 2, was One subsonic case and one supersonic case are
2.5 m long. This was to displace the outflow boundary shown as representative grid convergence trends.
away from the plate trailing edge. The first cell height of Figures 12 and 13 show total skin friction predictions
the baseline grid was varied according to the unit with inverse of total grid count for M = 0.4 and 1.2
Reynolds number as shown in Table 6. The first cell respectively, at several different unit Reynolds numbers.
height was fixed at both ends of the plate and exponen- Each computation was run out to establish solution con-
tially stretched from the surface to the outer boundary. vergence at each grid level so that total drag varied less
The upper boundary was 20 m away and the lateral than 0.00005 for several hundred iterations. Addition-
width of the grid was 0.098 m. ally, the difference in total skin friction coefficient
between the medium and fine density meshes was
The grids had the following dimensions. within 0.00005 for all unit Reynolds numbers except 1
milliodft. where the two levels were within 0.00008 at
Table 6. Reynolds number variation of grid spacing at M = 0.4. Slightly better grid convergence was obtained
surface at M = 1.2. These variations within the same bounds of
error are documented for the incompressible calcula-
Reynolds Initial grid
tion~.~
number y1 Yt stretching
(milliodft.) (lo-6m) rate Transition to Turbulent Flow
1. 7.50 0.42 14%
As mentioned earlier, explicit tripping was placed
2. 3.20 0.34 15%
at the leading edge of the flat plate. Transition to
4. 1.80 0.37 16% turbulent flow occurred at different locations down-
8. 0.94 0.37 17% stream depending upon the freestream conditions. The
15. 0.50 0.35 18% point at which the flow actually transitioned was deter-
mined for each solution by first calculating the peak of
30. 0.25 0.34 19%
the ratio of turbulent viscosity to the local bulk viscosity
10
American Institute of Aeronautics and Astronautics
