Boundary Lyer Theory

XXV. 1)obrlninalion of proRlo drag
included the solidity ratio 111 (= 0.6, 0.76, 1.0 and 1.25); the blade angle was Ps =
30" (turbine cascade). The loss coefficient defined in eqn. (25.33) is seen plotted in
terms of the dcflcxion cocfficient or deflcxion ratio
ad = Awd/?l~,, ,
where AN?, tlcnolcs 1.h~ tm.r~svcrsc compor~cnt of vclocit,y (i.'e. vclocit,y in circomferential
direction) created by the cascade. If we first center our attention on the
middle range of the polars (adhering boundary layers), we notice a steep increase in
the loss coefficient which occurs as thc solidity ratio decreases. The reason for it lies
in the fact that the number of bladcs per unit of length of the circumference is larger
when the pitch is small than when thc pitch is larger. To a first approximation the loss
coefficient is proportional to the number of blades. At the right and left edge of the
polar we observc a sudden and large increase in the loss coefficient. This is due to flow
separation on the pressure side (left end of curve) or on the suction side (right end of
polar) of the bladc. In the latter case, an increase in the flow angle causes the admissible
load on the blade to be exceeded. It is remarkable that the polar curves displace
themselves in the direction of largcr angles of deflexion as the solidity ratio decreases.
The ~ncasurcmcnts a.nd the calculat.ions were carried out for a Reynolds number
R -. to, llv - 5 x 10% .TIC calculations wcrc performed on tho assumption that thc
bountlnry hycr was turhulcnt. all along the hlatlcs. In the oxpcrin~ontal nrmngcrncnt
the boonrlary layers wcrc made turbulent by the provision of tripping wires ncar
the leading edges. Thc calculated and measured values of the loss coefficient show
very good agrccmcnt with cnch other. Furthcr examples and comparisons between
theory and experiment are givcn in [47, 631.
Wake: A very dctailcd experimental investigation of the flow in a turbulent
wake bchind a cascade of blades is described in a paper by R. Raj and B. Laksh-
minarayana [42]. Measurements included determinations of the velocity distribution,
intensity of turbulencc, and of the apparent Reynolds stresses in the wake at different
distances from the cascade. It has transpired that the wakes are not symmetric up
to a distance (314) 1 bchind the blades in cascades which turn the flow. The decrease
in velocity downstream from the cascade exit section is considerably slower than at
a flat plate, behind a circular cylinder or downstream from a single aerofoil at zero
incidcnce.
Jet flnp: The angment,at,ion of the turning angle A,'? = Dl - p2 of colnpressor
cascades by a jet flap has been investigated by U. Stark,[G4a].
2. 111flue11ce of Reynolds number: Thc chnngcs in the aerodynamic coefficients of
n cascadc protlucctl by a chnngc in l,hc itcynol(1s nnmber arc import,n~~l whcn it
becomes ncccssary to apply the results of tcsts on models to thc design of a fullscale
turho~nachine. This effcct is excrtetl principally on the loss coefficicr~t, and
the can be found discussed in a sizeable umber of publications 15, 41, 651.
From tho physical point of view, the cffect of Rey A" olds number on the loss coefficient
of a two-tlimcnsional cascndc is analogous to that of the skin friction of a single aerofoil:
hccausc in eit.her case the cffect originates in the boundary layer. The losses
sufTcred by the cascatlo stem mainly from the boundary layer if the pressure distribut,ion
dong a I)latlo in n cascadc is such that no imporhr~t scparntions occur.
'l'hry nro t,hcn ak:rl.otl hy tllc Ttcynoltls number in aho~lt t,ho same way as t,he skin-
I
friction coefficient of a flat plate at zero incidence arid are proportional to R-'12 for
laminar flow, becoming proportional to R-lI5 in turbulent flow. In both cases, the
Reynolds number is formcd with the blade length, I. Thc dcpcndcnccof t.hc loss cocffi-
cient on Reynolds number in the absence of separation can be determined by calcu-
lation with the aid of a method proposed by K. Gersten [15]. A rcsult of this kind is
seen displayed in Fig. 25.9. The diagram describes thc variation in the loss coefficient,
A g
(12 = -
;be4 ' (25.84)
of a cascade consisting of thick, strongly cambered bladcs, over a considcrablo range
of Reynolds numbers, that is from Rz = wzllv = 4 x lo4 to 4 x 105. I-Icre Ag dcnotes
the loss in stagnation pressure and iuz is the exit velocity. In order to providc a
comparison with mensurcmenta, thc diagram contains a thcorctical crrrvc which tnkcs
into account separation losses computed with the aid of Ref. [GO]. As far as the position
of the point of transition is conccmed, the calculation was based on the expcrimentally
verified circumstance that the boundary layer on the pressure side of a
blade remained laminar as far as the tmiling edge, whcreaa that on the suction side
undcrwcnt transition at the point of minimum pressure. Thc dingmm in Pig. 26.9
demonstmtcs t.hnh thcrc cxist.~ cxccllcnt ngrccrncnt I~cLwccn oalcnlntiotl nnql rtltrclrrilrament.
The magnitude of the losses is strongly influenced by t.he position of the point of
transition. As thc Rcynolds numbcr is incrcnscd, tho point of transition movcs for-
ward and this lengthcns the turbulent portion of the boundary layer and causes the
losses to increase. The forward movemcnt of the point of transition is enhanced by
increased roughness 1131 or by an increased turbulencc intensity [a], as one would
expect to find in a turbomachine. At very low Rcynolds numbers the boundary laycr
can separate before transition has occurred in it thus causing a large incrcasc in thc
Fig. 25.0. Loss coefficient of a turbine cascade, eqll. (25.34). in brr~rs of the Rryl~oltls tl~it~~bcr Rz,
after I