Boundary Lyer Theory

FRO XXII. Tho incornprcs~iblo turbulent boundary layer 11. Tho calculaLion of two-climcnsional turbulent boundary lnycrs (iX I
tribution U(x). Since the valuc of the shape factor, N(x) = 1, has already been
assigned, the only quantity that we need to calculate is the local Reynolds number
determined by eqn. (22.11a). Since m = const, we can contract the two terms on
the Icft-hand side and solve the problcm by performing two integrations, one each
for Rz(x) and R3(x). In ~ont~racted form these are
The relations and numerical values to be used for the exponents i, e and n, as
well M for the modified lrinematie viscosity v' are listed in Table 22.2, separately
for the momentum and for the energy method. The constant of integration is
DifTercntiat.ing eqn. (22.17) wit11 rc~pcctr to x and taking into account thc contmctions
dcfincd in Tablrs 22.1 antl 22.2 wc can tlcmonstrat~ eonsist~ency with eqn. (22.11a).
In the case of the moment,um m&hod, eqn. (22.17) becomes identical with eqn. (22.16)
if wc put i = n, R = Rz antl IZ = Ez.
Comparing the nurnerical data of Table 22.2, we find far-reaching agreement.
In spite of considerable difl'erences in the assumptions for the shape factor and for
the shearing stress at the wall we discover that thc two explicit cquat.ions for the
calculation of the momentum thickness are equivalent. The following specific numeri-
cal values can be recommcndctl:
The cnergy mcthod is discussed below.
Analogy with self-similar eolutions (ener~y method): Self-similar solutions in
boundary-layer thcory are generally described as equilibrium /lows when they occur
in turbulent motion. They are charnctcrized by the fact that the velocity profiles
u/U nt varying positions x become similar for certain velocity distributions U(x) of
the ontcr flow. 'J'his means that, the shape factor H(x) remains constant with z, that
is that dlfldx = 0. Figure 22.7 implics that all quantities which depend on x in
gcnrml n~ust bcrome constant for such equilibrium boundary layers.
We now suI)st,itutc in cqn. (22.10) the expression for cn from cqn. (22.813) and
note that the integration with respect to x can be performed in closed form with b =
const and p' = const. The Reynolds nnmbcr formcd with the energy thickness is
thus given by
. .
I hc numerical vn.lucs for 6, r, n, and v' are to Ip sclcctcd in accordance with the
relations in Table 22.2 and l'ig. 22.7s.
I
In the spceial rase of separation-prone flows for which Hs < I1 < Ilk, we find,
for example, that. 1 4- b$ = 1.004 and I + b, = 1.152. Tliesc two values differ by
about. 5%. Such a disrrcpancy can t)c tlisregardcd in view of the i~ncert~ainties in-
Iirrent in s~wh appro xi mat,^ mctzhods. In othcr words, this signifies that it is possiblc
to perform calculations using numerical values based on t,hc flat-plat,c ani~logy. 'rhc
quantity P' = PIPQ) that appears in eqn. (22.19) also depends only wcaltly on the shape
factor; by way of approximation, we let it be P' = I. 'rhns, the calculabion of ~(IIc
cncrgy t,hickness with the aid of eqn. (22.10) can be bascci on tdlc followitig numerical
valucs :
R += R3: h = 0.152; e == 2.3 ; n = 3.3; r' = 78 v; P' =- 1 . (22.20)
With thrse assumption, eqn. (22.19) transforms into cqn. (22.17) bearing in mint1 that
i = b, R = R3, and /C = "3, as cxpcctcd.
Refcrence [114] diows t,hat eqn. (22.1 la) suffices by it,sclf to solvc the prnbleln
when the energy mct,llocI is used. 13y contrast, when t.hc nionlent.urn mctliotl is usctl,
t.he coupling between cqns. (22.1 la) and (22.11b) cannot hc disregardcdt. 'Hie Iatkr
Icads us t,o the trivial result that Rz = IIz3R3 in view of thc dcfnit.ion Ifzn -- 02/fi3.
At. t.hr icvcl of approximation - -
considcrcd so far, thc momcntum mctliotl t.urns out
t,o be itlcntical witJr t.11~ cncrgy mct.liot1. Ncvcrthclcsrr, tho two proc:rtl~~~.c,s tlilli,~,
csscntially from one another in that tlic momentum mcthod employs tho t,wo basic
cquat,ions (22.11~~) and (22.1 lb), whcrras t,hc cncrgy mct,hotl gct,s by with cqn. (22.1 la)
nlonc. As far as tlic devclop~nent of furthcr approximat.ions in the for111 of simple
inkgrals is concerned, we have cxhaust,ctl t.hc pot,ent.inl inlierent in thc rnomrnt.um
mct,horl. In the energy mcthod, cqn. (22.1 1b) is used to tlcrivc n forninla for t,hc shape
fnclor by closed-form int.rgrnl.ion, as wc arc about, t.o how.
Integration method due to E. Truckenbrodt: 13. Trucltcnbrodt [I 11, 1141 worlrcd
out an npproximnt,~ mc:l,hotl for 1.11~ cxplicil inttcyrnl.iol~ of t,llc cqi~n.Iio~is of t.111.1~111v1it~
boundary layers which serves to obtain the boundary-layer t,l~iclrness (cnergy tliickness)
as well as the (modified) shape fact,or. The first vcrsion of the method [Ill] has
proved to be practicable for calculat,ions in engineering applications. It was, therefore,
tliougl~t useful to modify it in the light of more rccent discoveries. Employing
the classification introduced above, we tlescribc this as an energy method. Tlrc method
can be used for two-dimensional as wellasforaxiallysymmct~ricflows,c/. Sco. XXlX cl.
Tlic method is based on cqn. (22.11a) which is used to calculate the Iteynolds
number formed with the energy thickness. Hence we put R = R3, m = 2 and (IS =
@a = (21~) /? Rab in accordance with Table 22.1. If we assume 1,hat 6 = const and
p(r) is known, we can integrate eqn. (22.11a) with rcspcct to x and obtain
1:
1 B4x)
{R3(4I1+* = -;; [C1(2)1C where E3(x) = E3(x1) + JP' Un dx. (22.21)
The numerical values in cqn. (22.20) are valid up to P'(.T) = P(:c)/Pw. Ilowcvcr, an
inspection of Fig. 22.7b shows that /I' docs not clcviatc much from the valuc 1.0,
and we may calculate with /I' = 1.0 by way of approximation. In this case, eqn.
(22.21) transforms into cqn. (22.19). Thus, if no great dernands of accuracy are made
on the values of the Reynolds number, we obtain
--
z
2%
\ lllltbl
t By way of amplificstion, we mcnt,ion that the result, derived in Tal~le 6 of [I141 in also valid
when a -= const nnd b = ronut,.