The asymptotic formulations of the nonparallel linear stability of incompressible growing boundary layers are critically reviewed. These formulations can be divided into two approaches. The first approach combines a numerical method with either the method of multiple scales, or the method of averaging, of the Wentzel-Kramers-Brillouin (WKB) approximation; all these methods yield the same result. The second approach combined a multi-structure theory with the method of multiple scales. The first approach yields results that are in excellent agreement with all available experimental data, including the growth rates as well as the neutral stability curve. The derivation of the linear stability of the incompressible growing boundary layers is explained

Nayfeh, Ali H.

English

NASA Technical Reports Server

IN9 0 _ !2 514
NONPARALLEL STABILITY OF BOUNDARY LAYERS
Ali H. Nayfeh
Department of Engineering Science and Mechanics
Virginia Polytechnic Institute and State University
Blacksburg, Virginia
245
https://ntrs.nasa.gov/search.jsp?R=19900003198 2020-03-20T00:55:44+00:00Z
Nonparallel Stability of Boundary Layers
The asymptotic formulations of the nonparallel linear stability of
incompressible growing boundary layers are critically reviewed. These
formulations can be divided into two approaches. The first approach
combines a numerical method with either the method of multiple scales,
or the method of averaging, or the Wentzel-Kramers-Brillouin (WKB)
approximation; all these methods yield the same result. The second
approach combines a multi-structure theory with the method of multiple
scales. Proponents of the second approach have claimed that their
approach is rational and the first approach is not rational. The first
approach yields results that are in excellent agreement with all
available experimental data, including the growth rates as well as the
neutral stability curve. On the other hand, the second approach cannot
even yield the neutral curve for the Blasius Flow.
Introduction
This paper addresses the linear stability of incompressible growing
boundary layers. For two-dimensional mean flows, the streamwise
velocity component U(x,y) is a function of the transverse coordinate y
as well as the streamwise coordinate x. However, the rate of variation
of U with respect to x (i.e., aU/ax) is small compared with the rate of
variation of U with respect to y (i.e., aU/ay). Moreover, the transverse
velocity component V(x,y) is small compared with U and is a function of
y as well as x. For three-dimensional flows, the velocity components
U(x,y,z) and W(x,y,z) in the plane of the body are much larger than the
transverse velocity component V(x,y,z). Moreover, aU/ax, aU/az, aW/ax,
and aW/az are small compared with aU/ay and aW/ay.
To determine the linear stability of a three-dimensional mean flow,
we superimpose on it a small disturbance u(x,y,z,t) v(x,y,z,t),
w(x,y,z,t), and p(x,y,z,t). Substituting the total flow into the
Navier-Stokes equations, subtracting the mean-flow quantities, and
linearizing the resulting equations, we obtain
a__uu+ a_vv+ ?_w = 0 (I)
ax ay az
au au au BU ap 1 2
a_ + U _ + W--az + V--By + aX - -R v u
+ [u BU au BW
=o (2)
BV av av ap 1 2
_+ U_+ W--+az ay --vVR
+ [u aV aV + V av aVv :o (3)
246
I 2
Bw @w @w + v BW + @p _/w
+ u +w Bz -
BW _w BW
+ [uT + v : 0 (4)
where velocities, lengths, and time were made dimensionless using the
free-stream velocity U , a characteristic length a, and a characteristic
time a/U . Here, R = U_6/_ is the Reynolds number. The boundary
conditions are
u = v = w = 0 at y = 0 (5)
u, v, w, p + 0 as y ÷ _ (6)
The terms in the square brackets in Eqs. (2)-(4) are due to the growth
of the boundary layer (nonparallel terms).
Parallel Problem
Considering the parallel problem, one neglects the terms in square
brackets in Eqs. (2)-(4) and considers U and W to be functions of y
only. Then, one seeks a normal mode solution of the form
where
u : {I(Y)E, v = {3(Y)E, w : {s(Y)E, P = _4(Y) E (7)
E : exp[i(_x + Bz - rot)] (8)
Substituting Eqs. (7) and (8) into Eqs. (I)-(6) and neglecting the terms
in square brackets yields
ie_1 + Dk 3 + iB_ s = 0 (9)
1 (D2 2 2i(eU + BW - m)_1 + _DU + i_{ 4 - _ - a - 8 )_I = 0 (10)
1 2 2
i(aU + BW - m)_3 + D_4 - _ (D2 - _ - B )_3 = 0 (ii)
1 (D2 2 2i(_U + BW - m)_s + _3DW + iB{4 - _ - _ - B )k s = 0 (12)
_ = {3 = ks = 0 at y = 0 (13)
{n ÷ 0 as y ÷ - (14)
where D = B/By. For a given U(y) and W(y), Eqs. (9)-(14) constitute an
eigenvalue problem, which yields a dispersion relation of the form
247
,,, : m(_.B.R) (15)
A number of techniques have been developed for solving this eigenvalue
problem. These include shooting techniques, finite-difference methods,
Galerkin methods, and collocation techniques using Chebyshev or Jacobi
polynomials.
For the case of a two-dimensional mean flow and a two-dimensional
disturbance, W = O, us = 0 and B = O, and Eqs. (9)-(14) reduce to
i_{ I + D_ 3 : 0 (16)
1 (D 2 _ a2){ : 0i(_U - =);i + _3 DU + ia;. - E l (17)
1 D2 2
i(_U - _)_3 + D_, - _ ( - _ )_3 = 0 (18)
{i = _3 = 0 at y = 0 (19)
{n ÷ 0 as Y ÷ - (20)
Equations (16)-(20) can be combined to yield the Orr-Sommerfeld equation
(i:R)_I(D2 2)2
- {3 : (U - c)(D
subject to the boundary conditions
- _2)_ 3 - _3D2U
(21)
_3 = D{3 = 0 at y = 0 (22)
_3, D_3 + 0 as Y + _ (23)
where c = m/_. The neutral stability curve calculated using either Eqs.
(16)-(20) or Eqs. (21)-(23) are in good agreement with available
experimental data as shown in Figure I.
Recently, Smith I claimed the above methods to be "irrational" and
developed multi-structured theories for treating this problem. He used
a result from an "irrational theory" for the Blasius flow to observe
that "the typical wavelength of the n_4_rally stable modes on the lower
branch increases proportionally to Re z as Re ÷ _" and concluded _t
disturbances at the lo_ branch vary on a streamwise lenqth O(Re- z°)
and a time scale O(Re -'/") and hence they are governed by-a triple-deck
structure. Consequently, he let
x = I + _3X, t = E2T (24)
and
248
u,v,p=E = exp[ie(x) - inT]
where E = Re -I/8 and
2 3
n = =i + _=_ + E =+ + _ She =+L + "'"
de 2 3
dX
Then, he expanded the variables in the three decks as follows:
Main Deck
2 3
u = [u I + _u 2 + _ u3 + ¢ _nE U_L
+ ...]E
2 3 . ]
V = {eV 1 + E V 2 + E V 3 + e _nE V4L + ... E
2 3
__I_nl + _ P2 + _ P_ + E _,nc p_,,P + ...IE
where
y = E Y, Y = 0(1)
Lower Deck
2 3 ]
u = {U_ + _U 2 + E U 3 + ¢ _nc U,L + ... E
3 A+ 5
v : [_2V 1 + c V 2 + e V3 + E _nc V4L + ...]E
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
2 _ ...]Ep = [EP_ + c P2 + E3p3 + E _nc P4L +
where
5Zy : _ , Z = 0(i)
Upper Deck
2_ 3_ _ }E
u = [_ul + c u 2 + ¢ u_ + ¢ _nc U4L + "'"
2_ 3- 4 - ]
v = [_v_ + _ v 2 + _ v_ + ¢ _,n_ V,,L + ... E
2- 3_ h.
p = [_ + _ p2 + _ p_ + _ _,n_ P',L + ...]E
where
3-
y : ¢ y, _ = 0(i)
(33)
(34)
(35)
(36)
249
To account for the nonparallel effects, Smith had to include the next
term in each of the expansions in Eqs. (26)-(36).
Substituting the above expansions into the parallel part of the
disturbance equations (i)-(4) and boundary conditions (5) and (6),
dropping the terms in square brackets, putting W = O, separating
coefficients of like powers of _, and solving the resulting 36
equations, one obtains expressions for u, v, and p in the different
decks. Matching the resulting expressions provides asymptotic
expansions for u, v, and p. For the neutral stability curve, Smith
obtained
_3/2 %
=mn = 0.995 R a [i + 1.597 R- % + 10.02Fn R- a
+ 0.988 R 6 tn R6 + ...]
where
Ra = 1.7208/xRe
This expression is in fair agreement with the lower branch of the
neutral stability curve for large Re. However, its accuracy
deteriorates as Re decreases. In fact, it does not predict a minimum
critical Reynolds number.
Bodonyi and Smith 2 inspected the results of the "irrational theory"
to observe that "the stability properties of the Blasius boundary layer
are g_rned by the behavior on the streamwise length scale
O(Re -_z°v) as far as the upper branch of the n_w_ral curve is
concerned". Consequently, they used o = Re-,z-v as their perturbation
parameter and used the streamwise scale X defined by x = i + o X and the
time scale t = o_T. This choice leads to a five-zoned structure. To
account for the nonparallel effects, one needs to carry out the
expansion to 0(o ). In vlew of the logarithmic terms, one needs 13 terms
in the expansion. With three variables and five decks, one needs to
derive and solve 195 equations and then match the results. Bodonyi and
Smith gave up after four terms. Their calculated neutral stability
curve, which is intended to approximate the6upper branch, is below the
lower branch!! We note that for an Re = 10 , o = 0.5, which is not
small.
For the case of an accelerating=_ggndary layer, Smith and B9_gnyi 3
assumed a streamwise variation O(Re -_z-_) and a time scale O(Re -'z_)
near the upper branch of the neutral stability curve. Using this
streamwise variation leads to a five-z9D@Q structure, with the
nonparallel effects appearing at O(Re-_/'_).
It should be noted that the parallel flow assumption breaks _own
miserably for th_ case of Gortler instability. Floryan add Saric _ and
Ragab and Nayfeh J derived the appropriate equations for Gortler
instability for the:cases of zero and nonzero pressure gradients,
respectively. Hall u questioned the solution of the resulting equations
250
using a normal-mode approach and suggested solving them as an initial-
value problem.
Nonparallel Problem
A better agreement between the theoretical and experimental results
can b_ _tained by accounting for the influence of the nonparallel
terms - . To this end, we can use either the method _{ _xeraging or
the WKB approximation or the method of multiple scales _,*_. In this
paper, we use the method of averaging and let
u = A(x,z,t)_(y,x)e ie, v = A(x,z,t)_3(y,x)e ie (37)
p = A(x,z,t)_4(y,x)e ie, w = A(x,z,t)_s(Y,x)e ie (38)
where A is a slowly varying function of x and t,
ae _(x), De aeax aT = B, at
and the _n are given by Eqs.1{91_(14 ).
lengthy aTgebra, one obtains _-
aA aA BA
h 1 __ + h 2 _ + h 3 _- = h4A
where
(39)
After a straightforward but
hI = [ ({i_, + _s{3 + {s{s)dY
O
, * * * *
h2 : f [{t_4 + {4{I + U({1{t + _,{3 + {s{5)] dy
0
(4O)
(41)
(42)
. * * * *
O
(43)
a{ I . a_ s . a_4 . a{4 . a{ I .
h4 = - _ [Tx- _ + aT _4 + a_ _t + aT us + u (B--x-_i
O
a{t . at-- 3 . DE s .
+ a--_-_ + a--_- _ _ az az
aU + _;s BW)_ + (VD{+ ({_ _ + VD{_ _-
aV
+ r-'3DV + {3 -_
* aW aW_ *]dy+ _s a_)_ + (_ _ + VD_s + _s _J_s
where the _. are solutio_ of the adjoint homogeneous problem.
(40) can be rewritten as
(44)
Equation
251
where
aA aA aA
+ _ _ + _B --az= hAa_
h 2 h 3 h_
_ =h-_l' _B =Ell ' h = h_
Here, ,,, and _^ are the components of the group velocity in theQt . I_ ,
streamwlse dlrectlon.
(45)
Equation (45) describes the propagation of a wavepacket centered at
the frequency _r and the wavenumb_s or and Br, where the subscript r
stands for the Feal part. Nayfeh _ showed that for a physical problem,
_ and _B in Eq. (45) must be real.
For a monochromatic wave, aA/at = 0 and Eq. (45) reduces to
aA aA
--+ _ -- = hA (46)
ax 8 az
For a physical problem, Nayfeh 14 showed that w_/_ must be real For
. . ._ _theca eof aparallelmeanflow condo,,onreduc  to being
real, which was obtained by Nayfeh and Cebici and Stewartson using
the saddle-point method.
Two-Dimensional Mean Flows
For the case of a monochromatic wave, aA/at = 0 and Eq. (40) yields
A = Aoexp[ f (h4/h2)dx ] (47)
X
0
where Ao is a constant. Hence,
h4
u = Ao_(y,x)exp[i(_dx - _t) + f (_-_)dx] (48)
I
Consequently, the growth rate
o = Real [_x (In u)]
is given by
h.
+ Real [ ) + Real a
o : - _i _ [Tx (inc_)} (49)
The first term is the quasiparallel contribution, whereas the last two
terms are due to nonparallelism. It should be noted that the last term
produces a variaton in the growth rate across the boundary layer.
Since {_ is a function of y and, in general, distorts with
streamwise distance, one may term stable disturbances unstable or vice
versa. Morever, a different growth rate would be obtained if one
replaces u with another variable. For example, using v or p or w, one
obtains the growth rates
h4
o = - (1]. + Real (_) + Real[Txa (inCm] } (50)
252
where m = 3, 4, and 5, respectively. This raises the questions "What is
meant by stability of a boundary layer?" If the stability criterion is
based on o, then which o should be used? If one uses an N factor to
compare the stabilizing or destabilizing influences of certain
modifications to the boundary layer, then the contribution of the last
term will not be significant.
In the case of parallel flows, the last terms in Eqs. (49) and (50)
vanish and the growth rate is unique and independent of the variable
being used. Consequently, one can speak of neutral disturbances or
neutral stability curve given by the locus of _i(R,_) ._s0 However, inthe case of a nonparallel flow, the neutral stability Given
by o(R,m) = 0 and depends on the flow variable used to calculate the
growth rate and the distance from the wall. To compare the analytical
results with experimental data, one needs to make the calculations in
the same manner in which the measurements are taken. Available
experimental stability studies almost exclusively use hot-wire
aneomometers. Usually, they measure the rms value luI of the streamwise
velocity component u and use it to define the growth rate. Figure 2 2
compares the neutral stability curves calculated using lul a_Q +
with the experimental data of Kachanov, Kozlov and Levchenko I_. .Since
the experiment measured lul, the calculations of Saric and Nayfeh I0,
which were based on lul, are in better_agreement with the experimental
data than the calculations of Bouthier I, which were based on a- + V-.
Moreover, the growth rate is singular at the locations where luI = O.
Figure 2 shows also that the calculated locations of the singular growth
rates are in good agreement with the experimental results.
Some of the available experimental studies follow the maxima of lul
whereas others f_l_Rw a constant boundary-layer similarity variable q.
Saric and Nayfeh _,_v found that the contribution of the last terms in
Eqs. (49) and (50) are significant if one follows a constant n whereas
their contributions are negligible if one follows the maxima of lul,
yielding
h 4
o = - _i + Real(_-_) (51)
The neutral stability curve calculated by Saric and Nayfeh 9 using Eq.
(51), and shown in Figure 1, is in very good agreement with the
experimental data that Follow the maxima of lu|, except near the minimum
critical Reynolds number where the data may be suspect. However, in the
case of experiments conducted^by following trajectories of constant q
such as those of Ross et al. _u, the effect of the distortion of the
eigenfunction cancels the nonparallel effects, resulting in a better
agreement between their data and the results of quasiparallel theory.
Saric and Nayfeh I0 made other comparisons of the growth rates
calculated u_ng Eq. (51) with the experimental d_a of Strazisar, Prahl
and Reshotko _ and Kachanov, Kozlov and Levchenko I_. Strazisar et al.
* Also, private communication, June 1976.
253
conducted their experiments in a water tunnel and performed their
measurements at the maxima of lul, thereby minimizing the effects of the
distortion of the eigenfunction. They measured the amplification rate
as a function of the frequency at different locations on the plate,
corresponding to different Reynolds numbers. Figure 3 shows a good
agreement between the theoretical and experimental results. Kachanov et
al. also followed the maxima of lul and measured the amplification
factor a = lul/luol, where luo I is the rms value of u at the first
neutral point. Figure 4 shows a good _reement between the theoretical
results calculated by Saric and Nayfeh TM using Eq. (51) and the
experimental results.
T_ present nonparallel analysis was extended by EI-Hady and
Nayfeh_ to the case of two-dimensional compressible boundary layers, by
Nayfeh i_ to the case of t_ee-dimensional compressible boundary layers,
and by Nayfeh and EI-Hady L_ and Asrar and Nayfeh _ to heated boundary
layers.
Acknowledgment
This work was supported by the Office of Naval Research under
Contract No. NOOO14-85-K-O011, NR 4325201.
254
i.
e
.
o
.
References
Smith, F. T.: Proceedings of the Royal Society - London, vol.
A336, 1979, p. 91.
Bodonyi, R. J.; and Smith, F. T.: Proceedings of the Royal
Society - London, vol. A375, 1981, p. 65.
Smith, F. T.; and Bodonyi, R. J.: Journal of Fluid Mechanics,
vol. 118, 1982, p. 165.
Floryan, J. M.; and Saric, W. S.: AIAA Journal, vol. 20, 1982,
p. 316.
Ragab, R. A.; and Nayfeh, A. H.: Physics of Fluids, vol. 24,
1981, p. 1405.
.
7.
8.
9.
10.
11.
12.
13.
16.
17.
18.
Hall, P.: Journal of Fluid Mechanics, vol. 124, 1982, p. 475.
Bouthier, M.: Journal de Mecanique, vol. 12, 1973, p. 75.
Gaster, M.: Journal of Fluid Mechanics, vol. 66, 1974, p. 465.
Saric, W. S.; and Nayfeh, A. H.: Physics of Fluids, vol. 18,
1975, p. 945.
Saric, W. S.; and Nayfeh, A. H.: AGARD Conference Proceedings
No. 224, May 2-4, 1977, Laminar-Turbulent Transition, Paper 6.
Nayfeh, A. H.: Perturbation Methods, (Wiley-lnterscience, New
York, 1973).
Nayfeh, A. H.: Introduction to Perturbation Techniques, (Wiley-
Interscience, New York, 1981).
Nayfeh, A. H.; and Padhye, A. R.: AIAA Journal, vol. 17, 1979,
p. 1084.
Nayfeh, A. H.: AIAA Journal, vol. 18, 1980, p. 406.
Nayfeh, A. H.: International Union of Theoretical and Applied
Mechanics Symposium on Laminar-Turbulent Transition, (E. Eppler
and H. Fasel, Ed.), Springer-Verlag, Berlin, 1980, p. 201.
Nayfeh, A. H.; and Padyne, A. R.: Physics of Fluids, vol. 23,
1980, p. 241.
Padhye, A. R.; and Nayfeh, A. H.: AIAA Paper 81-1281, presented
at the AIAA 14th Fluid and Plasma Dynamics Conference, Pal,
Alto, CA, June 23-25, 1981.
Cebeci, T.; and Stewartson, K.: AIAA Journal, vol. 18, 1980, p.
398.
255
19.
20.
21.
22.
23.
24.
25.
26.
Kachanov, Yu. S.; Kozlov, V. V.; and Levchenko, V. Ya.: (In
Russian), Ucheniye Zapiski TASGI, VI, 1975, p. 137.
Ross, J. A.; Barnes, F. H.; Burns, J. G.; and Ross, M. A. S.:
Journal of Fluid Mechanics, vol. 43, 1970, p. 819.
Strazisar, A. J.; Reshotko, E.; and Prahl, J. M.: Journal of
Fluid Mechanics, vol. 83, 1975, p. 225.
EI-Hady, N. M.; and Nayfeh, A. H.: AIAA Paper No. 80-0277,
presented at the AIAA 18th Aerospace Sciences Meeting,
Pasadena, CA, January 14-16, 1980.
Nayfeh, A. H.; and EI-Hady, N. M.: Physics of Fluids, vol. 23,
1980, p. 10.
Asrar, W.; and Nayfeh, A. H.: Physics of Fluids, vol. 28, 1985,
p. 1263.
Schubauer, G. B.; and Skramstad, H. Ko: Journal of Research of
National Bureau of Standards, vol. 38, 1947, p. 251.
Wortmann, F. X: 50 Jahre Grenzschichtforchung, Braunschweig, F.
Vieweg & Sohn, 1955, p. 460.
256
420 -
360
300 -
x 240-
LL
0
400
o Schubauer & Skramstad
Ross _ ,.f
O Wortmann
"= 0 Strmzil•r otm/
0 Kmchmnov .¢
Parmllel
& Nayfeh
600 800 1000 1200
REYNOLDS NUMBER R*
Figure I. Neutral stability curve for Blasius boundary layer. Solid
symbols are Branch I experimental points. Open symbols are
Branch II. The critical Reynolds number is 400 for
nonparallel calculations, 520 ?or parallel calculations
(Saric and Nayfeh9).
257
n6
i ! i _ i
'.)\
\ o • /
\\o • i/
F:200 x 10 "_ a_\
•
,,,dP" 1 _ 1 _ I
300 400
Figure 2. Vertical variation of neutral stability points at F =
200xi0 -6. Experimental points from Kachanov, Kozlov and
Levchenko 17. Dashed lines are calculations of Bouthier 7
based on energy. Solid lines are calculations of Saric and
Nayfeh 10 based on luI. Streamwise position is the Reynolds
number based on ar which is the _ of Refs. 7 and 17. Solid
triangles give the locus of luI = 0 and the broken line is
the calculation 10 for luJ = O.
258
/
I
/
/
I
/
/
/
/
/
/
/
/
120
Figure 3.
Eq,(51)
240 FxlO 6
4
0
180 120
Amplification rate as a function of Frequency:
Saric and Nayfeh; Experimental data of
and Reshotko Following maximum of lul.
180 F x 106
Theory of
Strazisar, Prahl
C
F:110 x 10"*
• _Nonparallel
/
/ /---_arallel
/ / x/
/ /
I
• ' /
,' /
/ ,,
• ,///
• i ./_"
600 1000
F : 86 x 10"*
e/ •
• //
• /t
I -
1000
m/_Nonpa rat Ie I
/II I
/o
" -'- --'', Parailel
/i///
/
/
/
//
1400 600 1400
R* R"
Figure 4. Amplification factor a as a function of streamwise position
at F = 110xi0 -6 and F = 86 x 10-6 . Experimental points from
Kachanov, Kozlov and Levchenko following maximum of lul and
nonparallel results of Saric and Nayfeh based on following
maximum of lul.
259



Nonparallel stability of boundary layers

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