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Title Some Remarks on the Flow in Hydraulic-Machines
Author(s) Tabushi, Keizo
Editor(s)
CitationBulletin of the Naniwa University. Series A, Engineering and natural scie
nce. 1955, 3, p.9-19
Issue Date 1955-03-30
URL http://hdl.handle.net/10466/7671
Rights
9 N
Some Remarks on the Flow in IIydraulic-Machines
Keizo TABuSHI*
(Received January 19, 1955)
Abstract
With a view to obtain a rough estimation of the fiuid metion in hydraulic-machines,
equations in curved orthogonal co-ordinates (m, n, e) are given, taking the stream-line
on the rheridian section as m-axis, and some approximate relations ofthe flow are investi-
gated. Iri the first part are'described equations for the absolute flow in general (m, n,
e) co-ordinates and then in specia! (m, n, e) co-ordinates with some remarks. In the
second part,-the flow through runners is discussed. In the last part, some remarks on ,
the direct and inverse problems are given. .
1. Introduetion
For the analytical investigation of the flow in hydraulic-machines, especially in
runners, cylindrical co・ordinates or generally co-ordinates using parallel conical surfaces
are preferable. They are, however, inconvenient for the rough presumption of the fiow
and the quick grasp of the results obtained because the equations become very com-
plicated for the three-dimensional flow.
On the other hand, the curved orthogonal co-ordinates, taking a stream-line on the
meridian section as an axis, appear better suited for the approximate estimation of the
flow in hydraulic-machines and the design of three-dimensional runners, so investigations
are made on some properties of the flow by these co-ordinates. ' ' ' 2. Equations for the absolute fio.w in curved orthegonal codinates
' As showfi in Fig. 1, the zaxis Qf symmetry is taken as
2-axis, two curves on the me- -r
ridian plane intersecting at
right angles at any pointPare P
taken as m- and n-axes and the
circumferential direction th-
rough P is taken as e-axis,
thus curved orthogonal co-
ordinates (m, n, e) with P as
origin is determined. The cylindrical
When the external mass force has
equation holds
ct
B
A
. d
co-ordinates
the
.* Department of Mechanical Engineering, College
za ,n ' d" 1af Bx m R'"..,A' e .B.. 'c;'7 iiiile %.''E"c
C ix' A ,.t'..lll)! 211I' ;¥;l `eN ,g,
li:, B' IZ: .R 'B,
n ' Fig. 1. of P is expressed by (2, ag e).
potential 2, the following wellknown vector
'
of Engineering.
10 . Keizo TABusHi :-Ct--cxg :-grad(g+g+-S2 -vrote ・・・・・・・・・・・・(i) )
where c denotes the absolute velocity of the flow, P the pressure, p the density, v the
kinematic viscosity, e the vorticity of the absolute flow. If the unit vectors in m, n and
e directions are expressed by ii, i2, and i3 respectively, then
} = rot b = ii6m+i2en+i3ee
e.=-i,{8.(rce)-OoCe"}, e.==-IL{OoCom-O(ar;ie)}, 6e=g;ii"-aoC.m ・・・・・・・・・・・・(2)
Therefore, the equations of motion in m, n, e directions are
' atCm/ +c. OaC.m+ce 9gr-c. 3£i" -ce to.te-tt i2 sina : - o-O.(-il- +g+S2 )+yKin ・ "'''''''(3)
%C-t"-+c.3iZ"+ce %"e-c.{lit);nm-ce 31/ee-fr'}2 cosa= - -oOsii(e+2+!li2-)+vKh ・ ・・・・・・・・・(4)
ao-C-te + c.aO-jl'i +c. gl/ee - c.O-rSz - c. 9-S-"o +g!t;lti- sin cx + CerC" cos a =: - rge(-{} + 2 + C-22 ) +yKb .
-・-・-・・-(5)
.where a is the angle between 2- and m-axes and
sina=toi, cosa= 3: ・・・・・・・・・・・・(6)
K;.-72c.-3frCiy-S'","gllmi-90;`V3i£h'-,9.(31'£.")-,g,(3-ji';)-Si",a9-,Cz--!ig'-O-,Sij/I"-.(7)
Kh=72c.-Oo.2C,"-COS,aaoC."-Si".aOoC.m-tjl.r(OoC.m)-roOe(31/4e)'CO;a9oCbe-C-,eOC,oO-S-o--gl.(s)
Kb = 72ce- 9,-o2Cie2 - -li- bOiiim(t-oCege) +Si; a %z +op iSml (Sinr a) + ce zS}i(COrS at)
-i,-8.(lld'o")+CO;"9S3 ・-・-・-・-(9)
72=(oOmt2+oOn22+r20o2e2.+Sinrao-Om+CO;azlltT) ・・・・・・・・・・・・(10)
(3), (4), (5) are also expressed as follows;-
' OoCmt +c.OoCfi+cn OoCnin+ce %z-V'2 sina = -oeOm(-{lm +2) +vK;n ............(11)
{lit;t" + cmZiil'i + cn tblt; + ce 9g"e - 3I2 cos cv = - o-On ({l- +g) + vKh ・・・・・・・・・・・・(12)
) tbltO+c.gty;z+c. 3ti}'+ce -Orlg/t+CeCrm sina+CO "cosa= -roOo(-;-+g +vKb ・'''・・・・''''(13)
The left hand sides of (11), (12), (13) can be expressed as Dc./Dt, Dc./Dt, Dc/Dt
respectively. The condition for continuity is
divi=O , ・・・・・・・・-・・・(14)
' or Ou(a'-liim)-+51./(rcn)+tha'e=o ''''''''''''(is)
Some Remarles on the Elow in HZ!,draulic-Mdchines 11
or
g,C.in+ZS"+%ee+C-.M sina+ fil'L cosa ==O ・・・・・・・・・・・・ a6)
Equations (11), (12) are obtained also from momentum equations in m, n directions
using (16) and the last term on the left hand side of each is introduced because the
the sides C and C' or e surfaces in Fig. 1 are not parallel each other. Equation (13) is
obtained from the equilibrium of the moment of momentum about 2 axis.
When the flow is turbulent, we denote the component velocity cke (a==m, or n, 0),
as ca=cao+cd', where ckeo=the time mean value of ca, czx'=the variable part df cd.
Equations (15) and (16) hold for both ca, and c.'. Time mean Values are.obtained by
(11), (12), (13) with some modifications, namely, on the left hand side the first term
dissappears and the rest does not change its form, but become the sum of two groups,
one is expressed witfi cao for ca, the other cd' for cd of original equations, and on the
right hand side each term takes the time mean value keeping the original form.
Similarly we can obtain 'from other equations the expressions for time mean values.
cin-Stream-Line taleen as m-Axis
For the study of the flow in hydraulic-machines, it is convenient to take the
absolute stream-line on the meridian surface (we will call it cin-stream-line for the pre-
sent) as m-axis. Then
c.-o, 3I"fsc.3//i, Z-;"fec.g:, 9Szfscin9,", ・・・・・・・・・・・・(i7)
The equations of motion expressed in terms of the normal stress a and the tangential
stress T are obtained from the equilibrium of forces on a volume element as shown in
DDC7=-3/i/l+p-i,{{2-l/o'.Vm)+O(roT."m)+OoTeem}--pi-[o.tb"li+aeSi".`V+T..Oo-".] ・・・・・・・・・(is)
Ditn = -lll/l+i{O(o'a.n)+O(or.Tmn)+OoTeen}--}-[aeCO.Sa-o.gtin-T..g--".] ・・・・・・・・・・・・(ig)
PD-Ctp = -9a9o+iYr{oOoae+O(roime)+P(.rorn"e)}+-l;l[Te.Si"ra- Te. ,cosr `u] ・・r・・・・・・・・(2o)
Terms in the last brackets on the right hand side are introduced from the condition
that any pair of surfaces A and A' (or m-surfaces), B and B' (or n-surfaces) or C and
C' in Fig. 1 are not parallel, and terms containing Oa/On, Oa/Om within them are the
effect of A, A' and B, B' respectively. .
In most cases, however, the change of a in e and n directions is not large and we
can use equations (2)・w(14), putting the relation (17) in them, as approximate expres-
sionsi in such cases. ' As n-surfaces are composed of cm-stream-lines, there is some movement of the fluid
through them when Ocr/OO is not zero, but the magnitude of this fiow is generally
negligible compared with those through A, A' or C, C' in Fig, 1 so the condition of con-
12 Keize TABusHitinuity is expressed approximately as follows;
o-O.(':1'Cm)+,oOo(,:1'ce) ==O ''''''''''''(21)
where 1' denotes the thickness of the volume element inndirection. From (21)astream-
function ip.' can be introduced. Thus
i'c.= 9oOen', f'6,= -gg.b' ・・・・・・・・・・・・(22)
3. Equations for the fiow through runners
General Cage
Let e = the direction of the rotation of runner,
to = the angular velecity of the runner, u=rw the velocity of rotation, tv=the
relative velocity. ・When tu is constant and the relative fiow is stationary, the variation of b in time inter-
val dt at any space fixed point is approximately equal to the difference of c between
two points which are udt=-rde apart in e direction at any instant in that time interval,
thus
--,a・t-TC-e 'udt = -to g// dt .
'When the relative fiow is not stationary, a term (Oth/Ot)dt is added to the above quantity.
Theaefore in general
tbl'=-tuoO-eCT+{li-:'t and -bxF=-abxe-'u-×4,
- ti ×e = - ti ×g-i,a(,'o$) + i,tu ll+e = -grad (uce) +tu 8// ・
From above conditions the equation Qf motion (1) is transformed as follows;
toWt -di ×e = -gr.ad(g+2+g2- uce) -v rot } ・・・-・・・・・・・・(23)
If the external mass force is the gravity force .
a=gh ・・・・・・・・・-・(24)
where h is the height in the direction opposite to g. From the volocity triangle
w2 u2 C2 ・・・・・・・-・・・・(25) 2 -UCe = -2---2L
arid putting
ng =r, i= e +h+ tWi-t",2 -・----・(26)
Equation (23) becomes
tbl'- ab xg = - grad (gl)-v rot5 ・ 'm m""' (27)
Some Remarfes on the ')Fleotv in fl3,draulic-imchines 13
Also foliowing relgtions exist
S == rot th+2di= t+2w-' ・・-・・・・・・・・・(28)
rot}=rott+A, A =iiAm+i,An+i3Ae "''"'"'''(29) Am =2caO(/ronoa)', A. =2tuO(CrOoSaa), Ae == -2w(O /j;lla+aCoOnS ev).
Let
2 P- P2" --- P ・・・・・・・・・・・・(30)
then, from (23), (25), (28), (29), (30) we have
21ti-cb×ij I -grad(iPff+g+!ll2L -yrott-(2di×to+yA-) ・・・・・・・・・・・・(31) )
If we neglect the term (2di×nd+yA- ), Equation (31) has a similar form as Equation (1),
or (31) can be obtained putting nd and P in Place of 'c and P in (1), When the visco-
sity or the change of a is negligible, (2di×nd+yA)fu2tu-×to represents the Coriolis ac-
celeration, and we can say that if (P-pu2/2) of the relative fiow in a rotating runner is
considered just as P of the fiow in the same runner at rest, the difference of the two
fiows is caused mainly by the Coriolis force.
wm-Stream-Line taleen as m-Axis
For the computation of the flow in a runner, it is convinient to take co-ordinate
axes rotating with the runner. In this case, the relative stream-line on the meridian
surface or zvm-streamLline is taken as m-axis and the direction of rotation as e. Equa-
tions of motion in m, n, e directions are derived from Equation (31) etc. Neglecting the
fiow through n surface of a volume elernent similar as in Fig. 1, the condition of conti-
o-O.(,:1'wm) + ,oOe(2:1'we) = O ''''''''''"(32)
'and this is satisfied by a stream functiQn ipn.
]'wm=9aipe", ]'we=-ao-¢mn-- ・・・・・・・-・・・・(33)
When n-surfaces are constructed from relative
stream-lines or relative paths passing through er'a. wm-tshter:al:}tieiinfleo;}n anY'defi%teeseMg:irdfia'acnesSgg'". e nj' -.;li at.- iSIIIi E;:r
dle) ig l N ,,,
tion, throughcemes identically equal to zero and intersection
curves of these surfaces and other meridian
sections do not generally coincide with wm-
stream-lines on these sections if Oa/rOe=t:O.
Such surfaces are importent when I or H of the
fiow varies in the n direction.
In Fig. 2, Pb"JFInt" represents a relative stream
.oP C-ll J)o
... -LkP・
er -." x XN R
n
Fig. 2
ag
ty
14 Keizo CABusHr line and the surface (PP;."Ph"Pb") is composed of such lines, PP;n, PbiP'are zvm-
stream-lines and surface (PJe,s P;t' Pe) is composed of wm-stream-lines.
.Then
・ ::Iilt == hadOff- ww.e. ,. From the geometrical relation of the projected figure of the surface (PP."jP?,"Pb")
(which is assumed ta be an element of a plane surface) on the meridian section, we
have
PePe"s fPRn" sin 6fsdin tan 6f frde tane
:. tane=tan6lldlil =-tan6wW-eM ・・''"''''''(34)
313 ugi/Y - :Ilam tan6sin (a-・6) ・・-・・:・・・・・・(3s)
If tvm/zve, a, Oa/OO and the initial value of 6 are known, the values of e, 6 can be deter-
mined from (34), (35). Above relations are also obtained graphically.
Flbw of ldeal Fluid
When v=O and Ow/Ot=O we have from Equation (27)
nd×6 =grad (gl) ・・・・・・・・・-・・(36)
Equation (36) means that both thand 6 are perpendicular to grad (gl) or l is constant
along the relative stream-line.
' If wm=l=O, we have from the n component of vectors of Equation (36)
OotV.m==wmll/in+we.Ooae+.-1.o-a.(gl)--l)-:I:O(a'T.O) 'm"'""'(37)
and this is a condition for the velocity distribution on the meridian section.
If gl is constant on the n-surface and this surface is composed of relative stream
lines, then grad (gl) becomes equal to i30(gl)10n and from (36) 6n=O, and this last
condition (which is the same with the potential flow) determines the velocity distribution
on the n-surface.
Potential Flow' ,
In'many cases the absolute flow through runners is considered to have started from
an irrotational condition, so it can be treated as a potential flow if no free vortices are
shed from wall surfaces. In this case 6= O, Z =-2di ; and if Odi/Ot=O, we have from (36)
grad (gl)= O, gl=const ・-・・・・・・・・・・(38)
If there exists any exchange of energy between the runner and the fluid, the total head
'Hof the flQw must be changed where ' H=p/r+h+c2/(2g) ・・・・・・-・・・・・(39) From (38), (26), (25),
.
thme Remarfes on the rvoto in Hb,draulic-Atftvchines 15
grad (gH)=grad (uce) ・・・・・・・・・・・・(40)
The same result can be obtained from (1) as follows;
Ob Ond Oj b7/ == 57t -tu brt
and from l=O, tuab/OO=grad (ucJ), moreover Ow/Ot =O, y=O, so Equation (1) takes the ' 'form (40).
In ordinary runners grad(gH)ikO, grad(ucE)=VO, O(rce)/am=l=O, therefore from
en= O, Otvm/OO =kO.
We can say from this last relation together with boundary conditions on blade sur-
faces that in runners P also varies in 0 direction and by this rotating pressure field, the
energy is transmitted from the runner to the flow or vice versa, and in this state the
rotating blade or the rotating vortex surface gives the change of (rce) etc, and becomes
the discontinuous surface of the pressure and the velocity field. The velocity distribu-
tion on the n-surface can be obtained from boundary conditions and 6n=O, the latter is
also expressed, assuming Ol'/rOestsO, Ol'/Omftrj04/an as follows ,
aorm2.di,n+,O,2oipon,+g$n(Sin.'a-tb;'i)fe2ojsinat. ・・・・・・・・・・・・(4i)
If Oa/rOe =O, the above relation holds for the whole domain of n-surface, but in
other cases it can be applied to a narrow annular area on the n-surface through any
(2, r) position. The velocity distribution on 0- and m-surfaces can be obtained from the
ttboundary conditions and the following relations
0-surface ;
6e == tv. 31g.2 - aoW.m == o ・・・・・・・`・・・・(42)
' t. m-surface :
' ・1・・g. = -li-{O (orie) -- w. ao--ae} =o ・・・・・・・・・・・・(43)
If there is no exchange of energy between the runner and the fiuid, H and (rce)
become constant, w. and we do not vary in e-direction, so the fiow is axisymmetric.
The. wm-stream-line in this case can be determined graphically or experimentally by
wellkhown methods, and this stream-line or its inclination a can be adopted as the first
approximatlon ln varlous.cases.
Comparing this fiow and the potential (absolute) fiow with energy exchange, Equa-
tion (42) remains the same but the stream-line on e-surface is not the same, for the
distance of streamlines and the value of wm is not the same.
' Stream-lines of both flows, however, become the same if in the potential flow with
energy exchange (rce)=constant in the n direction on every e-surface, because in this
case Oa/OO becomes zero from (43). On the contrary, the stream-line on the 6-surfece
may separate from the wall if (ree) distribution is not appropriate,
the original co-ordinate axes and
only take the change of a and wn
(if it reaches to a considerable
magnitude) into account.
When the flow has variable gl at
the friction of upstream canal etc., the
so we can proceed the same way as
mation of surfaces with constant gl'
needed. Now we make remarks on
Assuming .Oa/Oe vO, the flow on
plane surfaces such as (R, e) plane. or
e) co-ordinates and (R, e) polar
following relations;
dm clR rdO P Rde -
e
16 Keizo TABusHr 4. Remarks on the direct and inverse Problems of three dimentional runners
The direct problem or the problem for the estimation of the flow character of a given
runner and the inverse or design problem are not simple for three dimensiona! runners.
There are some ingenious methods') for solving these pr6blems adopting cylindrical co-
ordinates, but in rnany cases they are somewhat complicated and inconvenient for a rough
but quick estimation. For the first rough estimation of above problems, the (m, n, e) co-
ordinates seem to be better suited. A method by these co-ordinates is as follows;
we take firs.t the (m, n, e) co-ordinates of an axisymmetric potential flow, then assuming
the absolute flow has the velocity potential, require the flow (zvm, we or ¢n) on several
n-surfaces from conditions of boundaries, of the irrotation and the continuity. In this
process the transformation of n-surfaces upon plane surfaces may be needed. For the
purpose of a very rough estimation, the flow on the transformed plane surface may be
approximated as a two-dimensional fiow. The next step is to check the velocity distribu-
tion on some e-surfaces and alsothe quantity of flow asawhole (a) e-S"r3F"Ce (b) (R,e)-p{ane
and the condition (46), etc., as z
described below. For many en- r 7ngineering problems only above 2steps are suthcient and the (m, n, .P 3 ・1e) co-ordinates are very effective -5urf
for them. When more precise nresults are needed, the flow on n- nTS"if
'surfaces are again treated with (c) (S,¢)-planeimproved wm,we,]' , a, but in Fig. 3this case it is preferable to retain
P2
P
'R,
@
¢ t-st
the inlet of the runner in the n direction due to
absolute fiow on the n-surface remains irrotational,
described above and the computation of the defor-
is only necessary when more detai!ed results are i some of the above subjects.
the n-surface can be transformed conformally on
(S, ¢) plane as shown in Fig. 3. Between (m, n,
co-ordinates or (S,¢) rectangular co-ordinates exist
' ' ' dS lv. -- tvR - ws' -do' ib5 -'riiJle'ilJJdi' '''"''"'''(44)
Some Remarks on the rvow in lfydraulic-Atlachines 17
putting e/e =x, ¢/@=x', ¢/e =x"=xxl we have i
£til = exp {x Sr {!IIZ} , S- s, = xt' Sr dll! , fir, = exp {S -i,Si} . ・・・・・・・・・ (4s)
Sr means the integration along wm-stream-iine in the domain considered. Then Equation
(41) is transformed inte following forms ;
' 3'-k4+R-'gk'(i-f3-".)+.022oipe2=2to'・ ''''''''''''(46)
gk-g-.-r,,3/I3Ls,g+g,2$-2,,n, ・・・・・・・・・・・・(47)
' where ip=ipn/1'me, ]'me == Si7'dw/Si din
' of =w( xi5 )2(i.mi'e) sina
of' : tu(x-rt/)27.m7'esina = tu'R2(lfi,J,)2 =tot(Sl,)2
'S? means the integration alorig the stream-line covering th6 total part of the flow con-
sidered. Velocity components and relations between them are (if S, ¢ are considered
dimensionless) ;
tv.=(i'itl!e)rOoipe, tve=--(j'Mi.e)gilh/, tvR= Soipo, we=-lllRlt, tvs= 31$-},
wdi=-・2g--}, ..m.・=jM,.e"-.R-.W-g. ---・--(4s)
ggLm ., jmie x-'Le .. eqg ............(4g)
ws 7r wipwhere e means unit length.
When (acr/On)=O and tu'=const, Equation (46) represents the two dimensional flow
in a runner rotating with angular velocity of and whose absolute velocity has a velocity
potential. Similariy, if (aa/On)=O and of'= const, Equation (47) represents the general
relation of the stream function of two dimensional fiow whose absolute velocity has a
velocity potential. But, as tu" is a function of of, x' and R, the fiow in (R, e)-plane
and its transformed flow in (S, ¢)-plane can not become potential flows simultaneously.
Moreover, in the present case, the transformed runner vanes form a straight row on the
(S, ¢)-plane and the potential fiow through these vanes satisfies the condition that Oa/
On=O and tu"=O or a=O in Equation (47). a=O means an axial flow in (m, n,e) co-
ordinates. The magnitude of x, x' or x" can be chosen at any convenient value.
The condition of similarity of the velocity taingle on both (m, n, 0)-and (R, e)-surfaces
is x= sinev. The condition that the ratio of any corresponding length at sections 1 and
2 remains the same for both (m, n, e)- and (R, @)-surfaces is x ={log.(r2/ri)}/Si(cim/r).
The transformation on (R, @) plane serves for the rough estimation of the fiew and
the boundary stream-lines if the fiow is treated as two dimensional, ,and the transforma-
tion on (S, ¢) plane may be useful fQr the rough numerical calculation of ¢.
)18 Keizo TABusHi On the blade surface, following conditions hold. If the blade itse]f is considered .ttOheb.e reCfO.MrrPiO.Sgedt.OkVgO. rEi,C:glllil4ilt,h..gCOrMelPaOtnioenn,tSeE;,L.C", Ce 'n m, n, e directions respectively,
Cm = len(tVedm-Web)/v/le.2+ ke2 == (WedmZVeb) sin a',
Cn = {lee(ZVmd- Wmb) h lem(Wed-- ZVeb)}/i/'lee2 + k.2 == (WebmWed) COs B' mu (ZVmb- ZV.d) sin Bl
Ce = len(ZVmbhZVmd)//k.2+le.2 = (Wmb-' ZVmd) COS r'
where fem,kn,lee, are direction cosines of the normal to the surface, al, B', r' are the
angles between n,e,m axes and the section curves of the blade on m-, n-,e-surfaces
respectively, suMxes b, d mean the quantities at the back and front surfaces of the blade.
A 't'-`- Let in Fig. 4 io ao be the entrance profile of the blade, a' i' be a part of the blade
.t--x -Asection on a e-surface, io' i' and ao' a' represent blade sections on two n-surfaces,
then
Si・:, Cndsn-' S::, Cndsn =S,".,' c dsj ・・・・-・・・・-・・(50)
where dsn and clse are elementary lengths of blade sections z
on n- and O-surfaces respectively. There exist also condi- erk 'M
tions similar to those usually considered in two dimensional Z v
cases such as; relations between Pb, Pd, tob, z-vd and gl: the L v or
direction of te and the value of ¢ on blade surfaces;4 and g 9nd at blade tips, etc, The conditions for the fiow asawhole / Ci{ e
are ; the total quantity of flow at'any section obtained from
the velocity distribution must be constant; the torque ob- e d 'M ctained from the pressure distribution on blade surfaces and
that from the velocity distribution around the runner must
coincide ; the relation between the circulation around a blade
section on the n-surface and velocity distribution on the
same surface around the runner.
Direct and inverse problems are often solved by trans-
forming differential equations into difference equations and applying
or the relaxation method3)., For this purpose dimensionless form of
Runners can also be designed by assuming the mean value of
surface, and this value is given to the middle portion between two vanes
distribution is obtained by extending the domain of known quantities')`)
of the flow. In such cases the following relations may also be used
・ e. ==o, O(or.Cb) .. OoWom
'
the condition of continuity is from (16), (17),
' a(orece) = -- r2 {Oowmm tw.(si"r "t + 23 I)}
p' 6 A o, Td di b d
Bn Fig. 4
the matrix method
(47) may be used.
O(rce)/Om on any n-
and the velocity
uslng equatlons
..・・・・・・・-・・(51)
'
-・・・・・-・・・・・(52)
r
Some Remarks on the Elow in H3,dra"lic・Mbuchines 19
.If the wm distribution of the axisymmetric potential fiow is taken as the mean value of
tvm, the left hand side of (51) and the right hand side of (52) are known for the
middle portion between two vanes and the variation of zvm and (rco) in e direction can
be estimated at this portion. From (51), (52) we have a differential equation for wm,
' ・ Oo2.W,m +9i6Wom, +06W.m (A,+B)+w.(AtB,+ct) = o ・・・・・・・・・・・・ (s3)
where A' =Sinra +23t;I, B' =2Sirna, c' = OoAi.
The separation of fiow from wall surfaces may be caused by the inadequate form of
walls and the effect of the viscosity. The former can be expressed as the effect of an
inadequate ditribution of (rc) and in some cases may be estimated from the flow of
ideal fiuid, while the latter is computed from boundary layers. A rough idea of these
boundary layers is obtained from boundary layers on axisymmetric walls and rotating
canals for which Equation (31) may be of use.
References
1) C. H. Wu, Trans. ASME., 74, 1363 (!952).
2) C. H. Wu, NACA TECH. NOTE 2214 (1950).3) R. V. Southwell, "Relaxation Methods in Theoretical Physics", Clarendon Press. (Oxford).
4) KC. Ho, R. J. Moon, j. of Appl. Phys. 24, 1186 (1953).