Boundary Lyer Theory

XXV. Dctcrminntion of profile drag
a) rrensurc distribution for vnrioun Rry-
noltk numbero nt Mz -- 0.3
b) Loss cocfficirnt (12 from cqn. (25.34)
nrr n function of tho Reynolds numbcr Rz
Fig. 25.10. Aerodynamic coefficients of a turbine cnscnde its a function of the Itrynoltls number as
menm~red by H. Schlichting nnd A. Dns [52, 531
n) I'rcauure dist.ribut,ion for vnriorls Rey-
tlolds nulnberu nt MI = 0.7
I
b) 'IJous coerficieut. Ira from C~II. (25.34)
as a function or the Mach number Mz tor
various vnlura of the ltcynoldu number
vig. 25.1 1. Aplclrlynnnlir. rotffirirntn of n t,t~rl,inr wnrnclr its n funrtion of Mnvb nr~mhrr nn measu-
T& by 11. Schlicl~ting and A. I h [62,53]
c. Los~es in the flow tlwnlgh cnscndos 775
loss cocfficient under certain circumstances. This large increase in the loss coefficient
at low Reynolds numbers is illustrated in Fig. 25.10b which refers to a turbine cw-
cade. At larger Reynolds numbws, Rz = 5 x 105, the transition is spontaneous nnd
the losses are small. At moderate Rcynolds numbers, Rz = 1 x 105, thcrc is Inn~innr
separation followed by turbulent re-attachement. Thus under the boundary layer
there forms a so-called separation bubble and the loss coefficient increases consider-
ably. At very low Reynolds numbers, Rz = 0.5 x 105, the laminar layer separates
and stays separated to t,he end of the blade. The losses increase by a large amount
once more.
The details of the separation of the boundary layer are once again mirrored in
the pressure distributions plotted in Fig. 25.10a for three valucs of the Itcynolds
number. The extent of the scpnration bubble depends strongly on thc Reynolds
number and on the intensity of turbulence of the oncoming stream. See [8,20,28,37,
43, 57, 601, and the pnpcr by R. Kiock [30]. C/. W.B. Robcrts 1431.
In conjunction with our discussion of the cffect of thc Rcynolds number, it is
necessary to stress that under certain circumstances the surface roughness can have
a large influence on the losses. In addition to enhancing transition, roughness can
also directly increase the losses. This occurs whcn thc protubcrnnccs cxccrd n crrlnin
admissible value; see [3, 561.
3. Effect of Mach numher : The preccding results concerning the Loss coefficient of
cascades refer to incompressible flows (M < 0.3). The effect of compressibility can
be said to set in at M > 0.4. An example of this effect is shown in Fig. 25.11b. The
plot represents the loss coefficient for a cascade producing a small angle of turn in a
subsonic flow. The Mach number Mz is the independent variable and t,he thrce curves
refer to three different Reynolds numbers. The pressure distribution for M = 0.7,
Fig. 25.118, shows that at Rz = 4 x 105 the loss coefficient increases sharply as the
Mach number is increased. The sharp increase occurs as a rcsult of shock formation
in region8 where the local value of the velocity of sound, cp, crit, has bcen exceeded in
the flow. For the two lower Reynolds numbers, Rz = 1.0 x 105 and Rz = 2-0 x 105,
the pressure distribution points to a separated flow. The results displayed in Figs.
25.10 and 25.11 demonstrate that the Mach number exerts a deep influence on the
flow through cnacades in the range of Reynolds numbers from R = 104 to 105, in
addition to the large effect of the Reynolds number itself. The preccding measure-
ments were performed in the high-specd cascade wind tunnel in Brunswick [54] in
which the Reynolds number and the Mach number can be varied independently.
The diagram of Fig. 25.12 illust.mt,es the effect of the Mach number on the loss
coefficient of a cnscade that produces a large angle of turn in the flow. 'rhc cnscadc
was designed for incompressible flow. The loss coefficient remains nearly constant at
the value Ct2 M 0.03 up to M2 = 0.7; it increases sharply as the Mach number is
further increased. The reason for this behaviour is clear from Fig. 25.13 in which it
is possible to discern the existence of shock waves on the suction side of the blade.
These cause separation of the boundary layer.
The effect of the Mach number and of the turbulence intensity on the loss coef-
ficient of cascades has been studied in two theses presentcd to the Engineering
University at, Rraunschweig by J. Bahr [2] and 1%. ITcbbeI 1211, respectively. Rcfe-
rence [50] may also be consulted on this point.