Boundary Lyer Theory

070 XXII. Tlic incompreaaibto lnrl)~rlent bounclary layer b. The calculation of two-dimenaionnl turbulent boundary lnyera 077
As ttlie hnsic rqiia~f,ions for nromen,tum. thickness &(x) and for e?asrg?/ thickn,esa
03(2:), we obtaint :
and
rmpcct.ivcly. Ilere CT is the skin-frict,iorr cocfficicnt and c~ is thc.dissipntion cocfficicnt.
r 7
llrr prrretling two coeifiricrib rela.t,ctl to thc shearing stress depcnd st,rtingly on the-
ltcynolrls nurnbcr, R, arcording to eqn. (22.2), and on the shape factor 11 in conform-
ity with cqn. (22.4). 'l'hrr following power-laws for their description lrnve withst.oot1
the fcst of time:
I I
'l?lic cxprcssions contain the factors a([{) and P(N) which are kque functions of the
shape factor and a specified power of the local Reynolds number~ Rz or R3. Thc dingrams
in Figs. 22.7% and b represrnt the quantities n. and h as well as a' = n/nm and
p' = fi/j?rn together with am and /?, (denoting values for zero-gradient flow) as
f~lnrtions of II. The rrsprctive form~rlac are quotcd in the captions. Jt is seen that
/?' varirs slowly with 11, whcrcas n' assumes the vnlue af(ll = Ils) = 0 at scpnration
antl t>llcn inrrcascs fast with incrcasing If.
Equations (22.8a, b) are now substituted int,o eqns. (22.7n, b) and this lends 11s
to the modified forms of thc momcntum and energy equations for &(x) and 63(x),
Pig. 22.7. Shearing stresses in turbulent bonndary layer/corresponding to eqn. (22.8) after [33],
[I 161; a) exponent8 n and b in terms of If; b) fnctors a and P in terms of If
t The above eqnntiona ncglect t.hc effcct of the normal components 3 and of the tonsor
of lteynolds stressrs from oqn. (18.10). Among othcrs, references [85, 87) contain indications
on how to modify bliooe equnt.iona if this uimplification is not acceptable.
In order to complete the evaluation it is still necessary to know the ahape-laclor
/un,dion H(x). It is shown on p. 487 of [I 131 that eqns. (22.7n, b) lead to tho working
forms
for the cnlculation of H32(x) and of HZ3(x). The shape factors can now be cnlculated
either by the use of a coupled pair of cquations, namely (22.7a) antl (22.98) or (22.7 b)
and (22 9b) To distinguish bctwcen these two possibilities we speak of the morne~tl~~rn
method in the first casc and of thc energy method in the second case. In most. proccdr~res
use is made of the momentum method, whereas E. Truckcnbrodt [I 11, 1141 favours
the energy method. Thc lnttcr choice has bccn madc for two rcnsons: (a) The Icft-lland
side of eqn. (22.7b), unlike that of cqn. (22.7a), does not dcpcrld explicitly on tho
shape factor. Thus instead of eqn. (22.7 b) we can also write
(b) TIlc dissipntion fnc(,ol. c~ on thc right siclc or c(,n. (22.70) munt bc (::llc\~l:~(rcl rroin
eqn. (22.8b) by performing a quadrature extended ovcr the boundary-layer tlricltness
0 ( y < &x), whercas the skin-friction coefficient c~ on the right-hand side of cqn.
(22.7~) tlcpcnds only on the local shcaring strcss at thc wnll, cqn. (22.411). 'l'llis
signifies that the dissipation work depends much less on the shape factor than thc
shearing stress at the wall. This is confirmed by the graphs of n'(I1) and P'(I1) in
Fig. 22.7b. Thus, in the energy method, the coupling between the equations determining
the boundary-layer thickness (energy equation) and the equntion which
determines thc shape fnctor turns out to bc much wcnlccr than in thc n~omcntum
method.
Reference [114] s11owsI1ow the basiccquntions for t,lre boundary-laycr t.llicltncsses,
eqns. (22.7a, h), can be transformed into equations that dctermine the local ltcynolds
numbers defined in eqn. (22.2). Similarly, this reference shows how the basic equations
for the shape factors, eqns. (22.!h, b), can be transformed into equations for
the modified ~hape factor defined in eqn. (22.4). In this mnnner, we obtain
Table 22.1 summarizes tl~c expression for the contractions m, di and y. The quantities
R, m, @ and p are provided with subscript 2 for the momentum mcbhod, and
wibh subscript 3 for the energy method.
4. Quadrature for the calculation of plane turbulet~t boundary layers. Under crrtain
simplifying assumptions it is possible still f~nrther to simplify the system of equations
(22 lla, b) In this manner it is possi1)le to derive explicit cxprcssions for R(n) and
N(r) by quadrature for an arbitrary velocity variation, TJ(x), in the outer flow, that