REPORT No. 62
TECHNISCHE HOGESCHOOL VLIEGTUIGBOUWK.UNDE Kemaalsttaat 10 - DELFT
2 5
FtB. 1953
THE COLLEGE OF AERONAUTICS
CRANFIELD
THE LAMINAR BOUNDARY LAYER IN SLIP FLOW
by
T. NONWEILER, B.Sc.
This Report rryust not be reproduced without the permission of the Principal of the College of Aeronautics.
VLIEGTUIGBOUWKUNDE Kanaalstraat 10 - DEUT
2 5 m.
1953
REPORT NO. 62 ?ï07E?,'iEER, 1952 T H E C O L L S G E O F A E R O N A U T I C S C R A N F I E L DThe Laj-ainar Boundary Layer in Slip Flow
-hy-T. Nonweiler, B.Sc.
SUl-imRY
This report discusses the effects of the existence of a small, hut finite, molecular mean-free-path on the steady air flow in the laminar boundary layer. The boundary conditions at the exposed s\;irface are modified by the e3d.stence of a slip velocity and temperature-jump between the air and the sijrface, and a theory of first-order approximation is developed to account for the con-sequent modification to the shear stress and heat flux: to the
s\3rface. The correction is obtained as a oi^antity of the order i_
of i^/3. -) coapared ivith the uncorrected values of the shear
stress and heat flux derived from the usual assimption of continutan flow (R being the local Reynolds nimber). The results are
applicable to a m d e range of external flow conditions, but are inapplicable to the internal conditions of the flow near the origin of the boundar-y layer.
In particirLar, it is found that, in the absence of .an external pressure gradient, the correction to the local skin friction coefficient is zero; and if, as well, the surface
temperature is uniform, then the heat transfer coefficient (lc„.) is reduced by an araount (lvl/3R ).
In ntnaerical work the results of this paper will be fotrnd significant if the local Reynolci^ number is between about 10'' and 10 but they can be used as a cruder approximation down
to about R ^ = 10.
A suriïïinry of the relevant physical theor;^- of non-uniform gases, and a comparison with other theoretical discussions of the proble-n of bovmdarji' layer slip flow, is included.
2
-CONTENTS L i s t of Syrabols
Introduction
The results of the Kinetic Theory of Gases,
Application to the Flow v/ithin a Boundary Layer
The Solution of the Boundary Layer Equations in Slip Flow
4. "1. General Conclusions
4. 2, The Solution for Zero Heat Transfer
4.3. The Correction in the Presence of Heat Transfer
4, 31. The Solution for Heat Treansfer in the Inconp]ressible Boundary Layer
4.32. An Extension to the Conpressible Boundary Layer with Heat Transfer.
4.4. The Accomraoda-tion and Momentum Transfer Coefficients
4.5. Limitations and Applicability of Solutions
Conparison with other Theoretical Results
Conclusions
References
Appendix Is The Physical Theory of Slip and the Temperature Jump
II: The Solution for Heat Transfer in the Inccm-pressible Boundary Layer
III; An Extension to the Conpressible Boundaiy Layer irith Heat Transfer
17: A Solution of the Boundary Layer Equations with Arbitrary Surface Temperatlire
Fig. 1, Diagrammatic Representation of the Method of Solution
LIST OP SB,rBOLS
A f i n i t e number
t .
B, coefficient of s in expression for (T - T* )/T
t •• ' w th^' a
C defined in eq-uation (ll. 3) of Appendix II.
C^ I c^ ds, the skin friction drag coefficient
C = (p, - p„)/2p„ u , the pressure coefficient p 0 a a a
5P incremental force
P(cr,t) defined in equation (ill. 6) of Appendix III, sjnd related to P(t) as sho^mi in (ill. 8)
F ( X ) function defined in equation (II.4) of Appendix 11, and evaluated in figure 2
G(x,y) value of an arbitrary property at the point (x,y)
G |x, y, T (x)i value of an arbitrary property of the boundar;'' layer at (x^y) if the air is slipping at the sxirfaxso
(y=0) where the temperature at a di. stance x from the nose is T (x) just within the surface.
G |x,y,T(xJ value of an arbitrarj"" property of the boundary layer at (x,y) if there is no slip of the air at the surfa.ce (j--^) v/here the gajs temperature at a distance x fran the nose is T(x).
K maximum value of Mo^
L cliaracteristic dimension of surface length
M = ' u /a , Ivlach number of free-stream a' a '
N number of molecules per unit volime
N(n) = s %
Nu Q L/k ( T * - T \ t h e N u s s e l t Number ^ s ' a^ t h vr»
Q r a t e of h e a t fIUDC
R = p u L / Q , sxirfac^ Re^rnolds number • " a a ' a '
R = p u x/|i , l o c a l Reynolds number 6 s si:irfaco a r e a element
S S u t h e r l a n d ' s c o n s t a n t , i n | i ö t T ^ / - / ( T + S ) T gas s t a t i c temperatixro
T,, s u r f a c e -uemperature f o r zero h e a t t r a n s f e r (the thermanetes
t h ' t e m p e r a t u r e )
-4-^ = vYP/p» "the speed of sound
c ' p e r s i s t a n c e f a c t o r ' ( t h e d e f i n i t i o n b y e q i m t i o n (4.15) i s assumed). /i 2 G^> = "^^/kp '>J'„7 t h e l o c a l s k i n f r i c t i o n c o e f f i c i e n t c gas s p e c i f i c h e a t a t c o n s t a n t pressiore P c ' ' I I I volume f t h e 'm.cmentum t r a n s f e r c o e f f i c i e n t ' k gas t h e i m a l c o n d v i c t i v i t y /i 3
\c„
= Q /i?p u , the local heat transfer coefficient
n s "^a a '
m molecular mass
n distance measured nomvi.l to surface
p gas static pressure
q gas mean speed
r dsfined in equation (4.25)
r(cr) = r(cr,0), the temperature recovery factor
r(cr,Ti) defined in equations (l7.14) and (17.15) of Appendix 17
r surface radius of curvature
c
s
x/t
t power of s in expansion of ( T - T,, )/T
u surface slip speed
VLfY
velocity components relative to surface along (x,y) axes
u',v' velocity ccmponents relative to plane of zero-slip,
along (x',y') axes
V mean molecular speed
x,y
system of orthogonal coordinates meas\ired along and
perpendi.cixLar to surface, with origin at nose of
surface
x'^y' system of orthogonal coordir^tes measured along and
perpendicular to plane of no-slip
£_i CjT = °f ~ ^f
A kjj = k.j - k ^
(HT) d e f i n e d b y e q u a t i o n ( l 7 . 1 6) of Appendix 17 m d e f i n e d i n e q u a t i o n (4^32)
a the accommodation coefficient
Y = ° r / ° v ' *^® r a t i o of t h e gas s p e c i f i c h e a t s
5 l e n g t h c l i a r a c t e r i s t i c of boundary l a y e r t h i c k n e s s 6 . t h e boundary l a y e r d i s p l a c e m e n t t h i c k n e s s
e v a l u e of s vrhere c „ exceeds t h e maxiraim v a l u e of c^, ^ t h e ' c o e f f i c i e n t of s l i p ' (see e q u a t i o n ( l . 7 ) of Appendix l ) ri = y / 5 , and d e f i n e d more p r e c i s e l y i n e q u a t i o n ( l 7 . 8 ) of Appendix 17 e = T / T a
?\ t h e g a s mean f r e e p a t h a t the svirface
[i t h e gas c o e f f i c i e n t of v i s c o s i t y
ë = (x-?o)A
g ,g' arbitrarily chosen values of x and x', respectively
p the gas density (= Nm)
cr = |ic /k, the Prandtl Number
T shea.r stress
"U. nonr-dimensional stream f u n c t i o n d e f i n e d i n eqviation ( r / . 3 ) of Appendix 17 Ü) = " ^ , assiansd a c o n s t a n t fj. dT ' S u b s c r i p t s ' a ' r e f e r s t o con>litions i n g a s i n t h e f r e e s t r e a m ' s ' r e f e r s t o c o n d i t i o n s a t t h e siu-face i n the g a s ' w ' r e f e r s t o c o n d i t i o n s on t h e s u r f a c e ' S ' r e f e r s t o c o n d i t i o n s a t t h e p l a n e of n o - s l i p (y - - ?(x))
'6 ' refers to conditions in gas outside boundary layer.
An asterisk denotes the val vie of a property derived on the assumption of continuum-flow (i.e. assuming A = O ) ,
A prime is used to denote differentiation of a
-6-1. Introduction
The first outv/ard manifestation of 'high-altitude effects' upon the structure of the boundary layer over a body is generally to be fovmd in the occurrence of a certain velocity of slip of the air at the surface of the body. It is an established fact that a viscous gas tends to adhere to any moving surface with which it is in contacts if the gas is regarded as a continuum then the assumption of '%ero slip' can be accepted. It is this tendency to drag the air along vd.th it v/hich caiises near a moving s^irface the appearance of a boxmd^ry layer of air, and within the boundary layer the viscous stresses are important.
If the gas is regarded, not as a haaogeneous medium (i.e. as a continv.vira), but as an aggre.gation of molecules, this concept of 'zero-slip' brcalcs down, since the gas cannot be said to be continuously in contact with the s^jrface. Likewise, the tem-perature of the gas at the wall is not necessarily that of the wall itself - there is a 'temperature jump'. The properties of
inddvidvial molecvtles striking the surface are theii* properties derived from their previous collision, wi.thin the main b'CLLk of the gas. In siinplified terras, we might conceive that the molecules possess the same properties within a distance of one mean free path from the sui-face. The implications of this concept are generalis?' unimportant for gases at ordinary temperature and pressures since the free path is very short. However, in the rarified air at higli altitudes of flight, the occurrence of a slip velocity at the S'jrface cannot be overlooked, as the mean free path may be appreciable carüpa;-red ivith the bovmdary layer thickness.
Even at lo\7 altitudes, it might be expected that the effects wou3.d not be neg3.igible near the origin of the boundary layer, vAiere the thiclaiess of the layer becomes very small.
This discussion concerns the correction to be applied to the estimated steady i-ate of hee.t transfer and skin friction to a tvro-damensional surface, on the assumption that the m.ean free path is small com.pared vrf-th the dimensions of the boundary layer, but not negligible»
Yfe shall first briefly state the physical Isiws governing the value of the slip velocity and the tempers-ture jump at a
surface - a more general discvission of the physics being given in the Appendix, for the benefit of those viho may be unawax-e of the approximations involved,; and then vre shall attempt to find haw fsj" these laws - which were derived for simple shearing motion and one-dinientional coir:/-ection of hes.t in a fluid at rest - may
be applicable to the flow in the boundary layer. After seme modification, these laws are sho\'m. to give a new set of boimdary conditions at the surface within the boundary layer, valid if the mean free path is small, so that all we have is a first approxi-mation. It should be noted that if the mean free path is not small but coiiiraensxirate m t h the boundary layer thickness, not only do these results break down, but also other ass\mptions inherent in the formation of the boundary layer equations become invalid.
Ye shall then show hov7 these modifications to the boundary conditions affect the solution of the boundary layer equations. In doing so we need only assijme that the flow is laminar near the surface, and vre do not need to cons timet the solution of the eqviationsj instead we relate it to other known solutions. These general restilts are then applied to some
particular solutions for the laminar bovindary layer, in a way which enables us to make certain quantitative deductions. A ccnpletely general set of results cannot be obtained, because of the difficulty of accovmting precisely for the effect of the surface temperat-ure j'ump. However, the results given cover a wide range of problems_; and a ccanparison with other theoretical insults is included.
2. The Results of the Kinetic Theory of Ga.ses
As is shown in the Appendix, based on the theories expanded in ref s. 1 and 2, in a one-dimensional shearing motion vri-th a gas velocity q parallel to a fixed v^all, physical theory leads one to expect that at the wall the molecules will possess a certain 'slip velocity'
^ = -(¥) ^^ • fco
relative to the wall, and are not (as is usually assumed) brotight to rest thereon. In this expression f is a factor called the'monentum transfer coefficient' vihich must be foimd from experiment, dq/'dn is the velocity gradient in the gas noimal to the wall
(meaned with respect to time), and A is the mean free path of
the molecules. The factor c is sometimes called the 'persistence factor' and in fact (c/\) me.y be related to the ustml definition of viscosity
H = c p V A (2.2) •
where p is the gas density, and v is the mean molecular speed.
Similai'ly if heat energy is being transported one-dimensionally through a gas at rest to a v/all, there will be a difference in the mean tempera.ture of the molecules at the surface
(T ) and that of the wall itself (T ) : this is a 'temperature jump' -vrfiich is usually neglected and is found to be equal to
•^3 w - Y+1 cr [^ajön^ - U . ^ ;
where a is called the 'a.ccoEmodation coefficient', cr is the Prandtl number (c u/k), and y is '''^he ratio of the gas specific heats (c^'^o )« The temperature gradient (dl/dn) normal to the wall is assigned to have a constant mean value within the gas, as - in the statement of (2.1) - was the velocity gradient. In fact, it is only necessary that these restrictions apply within di.stances of the order of A from the surface. At such dists-nces from a surface of a boundary layer very similar conditions exist, as WD.11 now be shown.
3« Application to the Flow within the Boundary Layer
Taking axes noiTial to, and parallel vri.th, a plane wall, the two-dimensioi^xL equations which govern the motion of a com-pressible boundary,' layer on this -wall are: that of continuity:
^ *
-iP
= 0
0.1)
t h o s e of consejr^/ation of momentum:
dn ÖV dp d / èn\ / , _\
0 - •"£ f^ ^)
and f i n a l l y t h a t of o o n s e r v a t r o n of energy, \7hich may be w r i t t e n a s •
/ 07 dl\ d / , dT\ fdv\^ dp / , ,v
c p ( u -,— + V -r— ) = T " ( k -r- j + u / -r- + u -r'^ . . . (3.4)
"^•p^ \^ ax a y / ay I ay / • \^dyj dx • • v^ ny
The symbols u s e d h e r e w5H be fo'ond d e f i n e d i n t h e l i s t a t the beginni:ig of thj-s r e p o r t .
These equations are obtai.ned using the usvial approxi-nation that quantities of the order of (5 '/L ) are negligible in comparison with unitj?" - where 6 and L are quantities
representative of the boundary layer thickness and storface length respectively^ and that the non-linear tenis in viscosity and conductivity may be ignored - an assimption which involves the neglect of terms of tlie order (|j.U/pL) compared Tn.th unity. As is ?rell~lcnown, both these assumptions are justifiable if R, the Reynolds mrabeij is large, and imply the neglect of terms of the
order (I/R) or (II'/R) compared with unity. Her-e we shall regard M as a quantity of order viniiy , so that in either case it is necessary only to regard R as large compared with unity to justify the assvimptions: tlie modifications otherwise imposed, if M is very small or very large, are discussed in refs. 3 and 4, but are unimportant to the principles vre wish to establish here.
Again, if the wall is curved and we measure x and y
along and perpendicular to the curved surface, then the
equations (3.1) - (3.4) still a.pply if we also neglect tenns of the order of (b/r ) compared with unity, where r indicates
c c the mean radius of curvature of the surface.
In the light of these approximations it is therefore pertinent to enquire if, say, the occurrence of a velocity of slip at the surface lias a^y significance TO.thin the accuracy of the usual approximaiiions. '7e have seen, in fact, from (2.1) that u is a quantity of the order of A X velocity gradient
s
normal to the surface: in the boundary layer at the surface, the nonaal velocity gradient is in fact a quantity of the order of u /5, the suffix 'a' denoting free stream conditions. Hence
^ = °(r) Ö-5)
From ( 2 . 2 ) , s i n c e v i s commensurate w i t h the speed of sovmd,
^=°(^) = °(l) (5.fl
vriioreas, a s a l r e a d y n o t e d
6 =
o{\) (3.7)
/ s o tliat . . .
Ï The statement that M = 0(l) here asserted does not of course
-lo-se that in (3.5)
(3.8)
In general, the value of u derived from the equations of the boundary layer has an inaccuraqy of the order of (u / R ) , S O that
it follows that, from (3.6), u is certainly significantly
s
d i f f e r e n t fran zero vri.thin the accuracy of the approximations.
Accordingly, we may vise the boundary condition u = u a t the
s
surface, instead of the usual condition of no-slip, without other-wise correcting the bovmdary layer equations: similarly we may
include the effect of the temperatvire jump. However, plainly it is impossible to do more than consider the first-order effects of these neviT boundary conditions, as second-order effects wovild not increase the accuracy of the solution simply because tlijere are other comparable effects TJhich are neglected as well,
To establish whether the results of the physical theory (given in para. 2) are relevant to conditions near the surface in the boundary layer, we note immediately that the variation of, for instance, the velocity gradient in the boundary layer has an
insignificant effect upon the value of the slip velocity, since such variations in the flow ivill introduce terms of correction to the expression (2.1) depending on (A/5); i,e, the proportionate variation in the properties within distances of the order of the mean free path. From (3.5) and (3.8), the value of (u /u ) is evidently a small quantity of the magnitude 1/R--, and so such collections will only affect the estimation of its value by mag-nitvides of the order of ( I / R ) : within the accuracy of üie boundary layer assumptions, such magnitudes are negligible. Hence, we may show that (2.1) is applicable to the conditions wlthin the boundary layer if \TO put:
u
.2cf'^) m
A (3.9)
s
(¥j (1)
In the formulation of (2,3) variations in, for example, the tem-peratvire gradient, have been neglected, but this is likemse justifiable within the limits of our approximations: however, no account has so far been taken of the mean kinetic energy of the gas relative to the wall. The kinetic energy of the gas near the 7ra.ll is a quantity of the order -^ u' per unit mass, whereas its excess heat energy is of the order c (T - T ) ; the ratio of these two quantities is, from (2.I) and (2.3)
2
% V? /c (T - T ) = 0 •=• s ^ p ^ s w^
The value of the velocity gradient is a magnitude of the order (u / 6 ) , and the value of the temperature gradient is in general
a
of order (T /5)*, so that the mean kinetic energy is a small a
fraction of the difference between the hea.t energy of the
P p incident molecules and the wall (since u /c T = (Y-1 ) M is in
a*^ p a ' ^ general of order unity), the fraction being a quantity of the magnitude (?^/S). Similarly the rate at which work, is done on the wall by the shearing stress is small compared vlth the rate of heat flux i.e.
u u s '^s
Thus the consideration of mean kinetic energy as \rell as of heat energy will not affect the formulation of the expression for the temperature jump, except in second order terms, v/-hich are
negligible to the order of approximation used here; so that we may v^rrite (2,3) S-s the condition governing the temperatvire jvmip at the sua'face in the boundary layer;
T - T = -k-X £
s w Y+l cr
m (f)/ ••••• ''•"°'
New suppose üiat v/e may express the variation of any property • (G) vd-thin the boundary layer near a point x c: t^ on the surface (y-OJ by a series:
G(x,y) == G(r,^,0) + (x-g^)G^(?^,0) + y<^.^M^,0) + ..,
Put (x-c^ )/L = g and y/8 = ri : then derivati-ves v/ith respect to 5 and T) are, in general, commensurate if (as is assumed) the quantity 6 / L = 0 ( I / R ^ ) is,small» Further let us consider the variation of the properties within distances of the order of a mean free path fx-om the point of the surface (,^ , 0 ) : suppose that x-.^ = Ah, y = PA , where A and (3 are finite. Then
G(x,y) = G(S^,0) + A (g! G^(?^,0) + P (^ 0^(5^,0) + ...
Thus wi.thin the accuracy of the solution, neglecting terms of order ( I / R ) ccmpared with those of order unity,
G(g^-f-AA,p;\) = G(CQ..O) + ^{^)%(Zo*Q) ' ^.(3.11)
For' example,
u O v ^ A , p A ) --= u^(c^) + pA. (||) j ....,...(3.12)
s
XsE^ o
-12-and from (3.9) in (3.12)
1
u ( V A A , P A ) = |_2c (2f£^ + pj A ( | ) j ...(3.13)
• x = ^
It follows that vteO if
y = PA = - 2c (^) A = - ^, say
(3.14)
The length S, often termed the 'coefficient of slip',
may therefore be envisaged as the distance below the surface where
the mean velocity of the gas would be reduced to rest relative to
the true surface. This apparent depression of the plane of 'zero
slip' ivill in fact vary with s = X / L , as the mean free path will
be related to the local gas temperature and the local wall
tem-perature. However this variation is small, and in fact negligible
over distances along the surface of the order A » as demonstrated
T^y (3.13).
At this plane of 'zero slip', the gas temperature is,
from (3.11),
T.
x=E,
= T ( 5 ^ 4 . A A , - S )
= T 3 ( ? J
-(a
'x=P
or, using (3.10) and (3.14),
( T ^
- T )
'x=^,
(3.15)
and evidently at the plane of zero slip, the gas temperature is
different frcsn the Tra.ll temperatvire.
is
Again at the plane of zero slip, the normal mass flow
= (pv)J + 0
'x=£ 'xt*£ v /
i,( \
-d(p u )
since, by (3.l), • ^
'•
at y=0 is equal to ~ - ^ ,
But there can be no flow into the surface at y=0, and so from
(3.5) and (3.6),
My_ rp^^J^'
a
^ - r ^ = i ) = o A =0^1^) (3.,.)
a a \ ^a a / ^ /
\^ ' J
l^cm (v/u ) i s i n general of order 1/fe- i n the boundary'- l a y e r
a
and so i t s value a t the plane of ' n o - s l i p ' i s zero, within the
assumed accuracy. Hence, at the plane of 'no-slip' the gas is sensibly at rest relative to the surface,
The concept of a plane of 'zero slip' has of course no physical significance, as the flow does not exist below the true surface; however it is justifiable as a mathematical notion, based on an analytic continuation of the variation of the mean
gas velocity deduced from the solution of the equations, and it is a notion which is a convenient one as it enables us to suggest a method of solution of the boundary layer equations in the
condition of slip-flow. This method v/ill be explained in the next paragraph.
Before embarking on these arguments, one fundamental point about this method, of solution needs clarification. In equation (3.1l), the property G is arbitrary: it might be the velocity u, say, or the shear stress, which involves -r— . Yet, if it refers to the velocity u, then as in (3.12) v/e assume that T/ithin distances of the order A frcm the surface the velocity varies linearly with distance: this implies that (with constant viscosity) the shear stress is constant over this distance. It is apparently, therefore, inccripatible to proceed as vre shall do, to calculate the change in shear stress by the use of equation
(3.11)» Ho-wever, there is no incompata.bility. To illustrate this, let us consider a simple example. As a first approxima-tion, equation (3.1l) tells us that the shear stress varies linearly with distance close to the surface: assuming say that thje viscosity is constant, then this admittedlj'' Implies a quadratic distribution of velocity. We have, however, that the shear stress at the true surface at a distance S from the surface of no-slip is
T^ = T^ + ^.S
S
/.2 \
Now, |i (~^) at the surface of no-slip is equal to the pressure gradient aïong the surface -^ (frcm the equation of m.otion of the boundary layer). Thus the change in the shear stress between the true surface and the apparent no-slip surface is .
''3 " '^S = ^ dx
which is a rcs-ult rre shall later derive again. To evaluate this difference we need to Icacrff ^ a first approximation to ^ may be obtained, as in equation (3.14) "by assuming a linear variation in velocity; thus even if i-ve malce the more acceptable assumption
_________^___^___ ; /of a . ., a i The author is indebted to llr. G.M. Lilley for suggesting this.
-14-of a quadratic variation in velocity, we find
that:-^ /auN 1 that:-^2 / dthat:-^u\
-3-2 * H'4/^-
(~^^^
i.e. from (3.9), since Vy = 0, (d-ü/dy)^ , and (a u,''a;r')^ are constant
/.2
!•» 6» ^ ~ ^ C j
(¥)^t; = (¥)^(^^i)—J
""7The first-order approxrination to "C is adequate, and there is therefore no incompatability in making the assumption of a linear variation in velocity which leads to this approximation. This assumption is sufficient to yield S.
4c The solution of the Boundary Layer Equations in. Slip Flovy
^^ ' Genoral_Conclusions
Suppose v.'e now introduce a system of or-thogonal
coordinates (x',y°), with the lines y' r: constant lying parallel to the plane of 'nc-sllp' (y = - ?(x)), and y' = 0 corresponding to the plane of 'no-slip'. The curvature of the lines y' = const,
p p
is then simply proportional to (d ^ d x ) since ^ is a small length, of the order A J their radius of cvirvature is thus a magnitude
.2 ' '•
r_ = ^ 0(1)
p -7 /p
SO that 5/r is of the order (6A/L ) i.e. of the order I/R ' . c
Hence, in the light of the remarks which we made at the beginning of the last section, it follows that if we neglect terms of order
(I/R) compared with unity, we may write do^,"m the equations governing the flow Yri-thd.n the bounda^ry la^^-er as
ik^'l , ikill
- 0 (4,1)
pu'-^-r+pv'^ = - ^ , + 3~,
[\^-^)
....<... (4,2)
c p
/ , ar ^ , aT^ , dp a / ,
Ö T \/au'N
P (:;* ^' ^ -' a^y = -• d?'/ a7' C^ aj-; ^ ^ ( ^ j
(4.4)
where u ' and v ' are the v e l o c i t y components p a r a l l e l t o the x ' and y ' axes. Except for the s u b s t i t u t i o n of primed symbols, these are i d e n t i c a l with equations ( 3 . l ) - ( 3 . 4 ) .
Now, i f - as i s visual - the equations ( 3 . l ) - (3.4) are solved subject t o the c o n d i t i o n of n o - s l i p a t the t r u e surface, the boundary conditions t o be imposed a r e :
u = V = 0 and T = T a t y a 0
(4.5) u = vu and T = T. a t y = O '
the boundary conditions and the solution bring valid only for
X ^ 0 where x=0 corresponds with the point of origin of the boundary layer. Yet, if T/e allow the existence of a slip-velocity and introduce the new (x',y') axes, these havtj been chosen so that the corresponding boundary conditions are:
u' = v' = 0, T = Ty , at y' = 0 (4.6a)
since from (3.16) we fovind that the gas is at rest relative to the surface at the plane of no-slip. At y = '•yO , corresponding to the condition u = Uo , we have that
/ / "
^ = (""-'i)/v-(i)
But ( d ^ d x ) = 0 ( A / L ) = 0 ( I / R ) , from b'S): so t h a t within t h e approximations of the s o l u t i o n , u ' s v u a t y s oO ^ which i s a l s o the condition a t y ' = c ^ . Hence we a l s o have t h a t
u ' = Ug , T =. Tg a t y' = o'y . . ( 4 . 6 b )
Without loss in generality, we may assert that these conditions
are valid for x'JS 0,
Comparing the equations (4.6) v/ith those of (4.5), it follows that the boundary conditions, as well as the actual
equations, are similar; whence we derive the following important result.
To the order of approximation implicit in the assumption of the usual boundary layer equations, in slip-flow the conditions at a point (x'jy'),-referred to axes parallel and perpendicular to the plane of no-slip (y' = 0 , or y -. -• g(x)), - are identical with
- 1 6 -
TECHNISCHE HOGESCHOOL
VLIEGTUIGBOUWKUNDEKanaalstraat 10 - DELFT
those in the-"bcninaary- layer over the same svirface at a point
(x,y), - referred to axes parallel and perpendicular to the surface (y=0) where a condition of no-slip applies; - except that, in the former problem, the temperature of the gas at the plane of no-slip is different frcm that in the latter problem, where the temperature of the gas at the surface is that of the wall (i.e. there is no temperature jump).
Let us attempt to formulate this condition mathematically. Suppose we denote by G tx,y,T(x)J a properiy at the point (x,y) in a flow without slip referred to axes parallel and perpendiculair to the surface (y=0) \7here the gas temperature is T ( X ) , and xs=0 is the origin of the plate; and by G l_x,y,T(x)j the same property at the point (x,y) if the air is slipping at tlie surface, where the wall temperature is T ( X ) . ' •>
y'=o,
whereNow, in our notation, a point on the plane of no-slip corresponds to the point y
x'=5;,
^o
- i^(E,^), and x=:^^ ,
•^F I—
1 + dx = i^
'riii
i.e. where E,' = E, , to the appropriate order of approximation, Hence, the conclusion already stated implies that
G Ky,T^(x)] = G* [x,y
+ ZU),
T^(x)]
(4.7)
In particular, the value of properties on the surface -where y=<D, are given byG !x,0,T^(x)J = G* [x, ?(x), T^(xTj
= G* [x,0,T^(x)] + S(x).G* [x,0,T^(x)]
(4.8) using the results of (3.11). It will be noticed that in the snail ccxrrection term multiplied with ^(x) it is immaterial, to the order of approximation valid, whether the quantity G is that derived for a gas surface temperature of T or T>^ , since
w «^ the difference is small.
However, it is the difference betivcen these two quantities which significjantly affects the value of the first tern on the right-hand side of (4.8). We require to find the difference
,/^G =
G [ X , 0 , T ^ ( X ) ]
- G* [x,0,T^/xj]
and t o do so, vre e v i d e n t l y need t o find the difference between
G i x,0,T I and G j x , 0 , T ^ | . "/e may require a knovdedge of
the former quantity - since thd.s i s the value of the property
derived far a continuum flow with zero surface s l i p and
tempera-t u r e jump; butempera-t tempera-t h i s i s of no value i n finding tempera-the l a tempera-t tempera-t e r , since
(T - Ty) v a r i e s m t h x.
This may be seen from oqviation (3.15), which gives the
value of (T^ T ) . Only i n the condition of zero h e a t t r a n s
-f e r . I -— i = 0, i s the e-f-fect o-f t h i s change i n temperatvire e a s i l y
\"^)Q ... . a
accounted for, since then the temperature jvmp i s zero.
Poiother conditioriS, i t i s not easy exactly to c a l c u l a t e tlie d i f f e r
-ence between G Ix, O-T^l p.nd G !x, O.T . Talce, for example,
the case of a f l a t p l a t e with a uniform surface temperature,
T = c o n s t . ; then A does not change with x, but {-r^A . and
so a l s o (T^ - T ) , v a r i e s i n general as l/v'x, and t t e r e i s no
u, w
general solution for the flow over a f l a t p l a t e -ivith a temperature
d i s t r i b u t i o n T^ = T + Bx '- (where B i s a constant) except a t
incompressible speeds of flow.
Po.ri;unai:ely, the conditions of zero heat t r a n s f e r and
low Mach nvTiober, are both important onesj and the effect of the
temperature j-jmp r.vjy otherwise be a s s e s s e d i n the p a r t i c u l a r
condition i n v/nich the shea:r s t r e s s i s ind.epend£nt of the
temperatvTi'e distribution^ t h i s i s so i n a compressible flew
without a pi-essure gradient i f the v i s c o s i t y v a r i e s i n proportion
t o the absolute temperature. Although t h i s l a t t e r i s a condition
not exactly s a t i s f i e d i t does give an adequate approximation t o
the r a t e of heat t r a n s f e r , and presumably a l s o t o the effect of
the temperature-jvcip, ovei' a f l a t p l a t e , provided the Mach
number i s not too h,:5«i!ji.
4 ' 2' The Solution for Zero Hcat~-Tran5fer
If the?re i s no lieat trfxnsfer to the svirface
\dy,'
and so frcKi (3.15)3
4 •«•
1 8
-Hence, equation (4.8) may be v;ritten as
G j^,0,T^^(x)| = G* [i,0,T^(x]l + ^(x) G* |^,0,T^(x3} . . . ( 4 . I I )
Consider naff the shear s t r e s s i n s l i p flow:
=
("t)
4s
Putting G = T;, we may in (4. II) immediately relate i;„ to t'
(i.e. the shear stress at the vra.ll, calculated for the same conditions with no slip at the surface.) In fact;
x , = < . S { x ) ( ^ ^ •• (^^2)
But frcm (3.2), if the condition of no-slip is imposed, at the
wall
(
'f)^=i t^-^"
so t h a t
*^ s s s dx
p
In terms of the locrl. sl<:in f r i c t i o n c o e f f i c i e n t , c^ = ''^„/?P„ii_ »
using the value of Z given i n (3.14),
, V , = 0 , - o* = 2= ( 2 ^ j ^[^) (4.14)
Now A i s the mean firee path of the molecules a t the v/all, where
the gas temperature i s T = T • and so from (2.2),
p^T ^ \ \
° ^ = ^^s/Ps^s = ^^a/Ps^a = i f T ^ ^ - f " ) ^^'^^^
a a
because*, -within the accuracy of the k i n e t i c theory' of gases
-"'-'• • - • ' • — •• — r
* The implication i s here t h a t Yre t r e a t [iccT^, which v/ould seem
t o preclude the consideration of other conditions of the temperature
v t i r i a t i o n of v i s c o s i t y which we aftervrards introchice. The point,
however, i s t h a t the v a r i a t i o n of |i cannot be accurately described
by ( 2 . 2 ) : i n S u t h e r l a n d ' s treatment of the problem, fcr i n s t a n c e ,
(see r e f . 1 , §90),
(1 + ly^ = o^ Ps^s^«^^t
where c. ha.s a t l e a s t a value close t o t h a t of c, and S i s
Sutherland's constant. Sutherland's formvila i s knoivn t o give a
more accurate d e s c r i p t i o n of observed data, and frcm i t we deduce
v/hich, a p a r t fran a difference i n the constant of proportion, i s
tlie sane as (4..I5). Thus we sliall regard (4.15) as d e f i n i t i v e , i n
preference t o ( 2 . 2 ) , whatever the v a r i a t i o n of [i with temperature,
as i t displays the e s s e n t i a l fact t h a t (cA ) v a r i e s i n v e r s e l y vTlth
d e r s i t y : the constant of propoi-tion i n (4.15) may i n f a c t be i n e r r o r .
If (4/15) i s antvnried,, then, t o be a definitio;.:., the assizr^ptions implf.cit
^ c c l / p and s o |j.tx V. The mean molecralar speed i s
^ = 2 ^ i = 2 / ^ a ..(4.16)
30 t h a t , c >\ / Y TT: _ S / ^a A / ü i ï ï'^ _2. ^^ A >i-7^" L '^ ^ 8 ' T " VP a LJ ^ V 8 R T p^ • " • ^^^'''
Si ^'^B. a. y a o Thus, from (4.17) i n ( 4 . 1 4 ) , s i n c e T = T , . , - \f )M' 2' R pg T ; \ d s /Ac.
f \X /M Z R PR ^ ^ ^ ' \ ( 4 . 1 8 ) ••# • .] dC \ 1 /,,2'>2or
R'= C^,= (R^^ „p ^ /2:i£j / Z i / V t h A ( ^ \ ( ^
In particular, we see that if the surface pressure gradient is zero then slipping has no effect on the skin friction; the correction in the presence of a velocity gradient becomes most important at high Ma.ch nvtmber.
Similarly, if
Q
is the rate of heat flux in the gas through the boundary layer,
Q
since, frcm (4.9), there is no heat flux into the v/all. Frcm
(3.4) .
<* p [.£&] a - Ll /•%') . . , . . . « ( ^ ^ 2 0 ) Vay/g ^ s ^ a y / ^ so t l m t i n ( 4 . 1 9 ) , from ( 4 . 2 0 ) , i f A Q = Q - Q* , t h e n •'s s „ 3 In terms of the local heat transfer coefficient, k„ = Q /%p u ,using (3.14)
A k - - k*
(^--^)
^-
-^^'
(c*)2
O IC^j - K^ - ^ V f / L ji ^°f ^ o r frcïTi (;+.17), i f \i/ii^ = ( T / T ^ ) , p u L S CO
-20-^^--m/^^K>)
1-03*\2
M(op
..(4.22)or
V
/ 2 • irR
J
This expression shoivs that the heat transfer is zero vAien in fact the uncorrected value of the heat transfer coefficient is positive: i.e. for small rates of heat transfer, the effect of surface slip is to reduce the rate of heat transfer, most particularly at high Kach number. The effect is quite independent of the surface pressure gradientc
Finally, we note tliat the displacement thickness of the boundary layer is reduced, by slipping, by the distance S, i.e. ^1 = = o u O * \ y [ / U
h
. ) ;
^ ) d y
-J
\o
-z
e-^)^^
But for - ^ < y < 0 , (u/u ) i s of order ( A / 5 ) so t h a t
or, cjorrect t o the f i r s t order of apprcoc3Jiiatioru!
6, . 6 ; - S = 6 t - 2 . ( 2 . ^ j h
Thus, frcm (4.17),
or
A 8 ,
2-f .V %
f >! 2
I /Pa^
M
R
-{
r 1 5^R'^
Mlf J ( ^ /
Ï_2L/V\h\l
i . . (^4^23)M3
RJ
4-13. The Ccyrection j . n the Pre_sejiGj3 of Heaj: Transfer
As alread3'- remarked, the s o l u t i o n of the problem of the
e f f e c t s of svrface s l i p and temperature j'-jmp on the flow of the
boundary layer, with heat t r a n s f e r to the surface, i s more complicated.
This is because in the analogy with the no-slip flow about a
surface depressed below the true surface, the temperature at this
surface of no-slip is not simply the v/all temperature, but differs
from it by a certain araount (varying v/ith x) as determined from
equation (3.15).
The study of this problem involves the solution of the
bovmdary layer equations of no-slip flow about a surface vath
varying surface temperature, and this problem only becomes
tractible if some simplifying assumptions are introduced. Such
additional assumptions are detailed below» even so, the results
(which are also stated and discussed below) are a little involved,
and their derivation is relagatsd to the Appendices II-IV.
4.31. The Solution for Heat Transfer in the Incompressible
Boundary Layer
The term 'incompressible' applied to a boundary layer
with heat transfer is rather inappropriate, but we mean by it the
conditions obtaining if the Mach number of flight is small. If
v/e further make the assumption that the change in temperature
across the boundary layer is small, then the energy equation (3.4)
becomes
aT aT / k N a^T /, „, s
u -r- + V T " = I —~* —
T^ •
• • • 14.24j
^-
^y \P^p/ ay2
where u,v are sensibly the velocities in the layer if the
temperature is every\7here constant, and j ~~~-J may be treated as
a constant. A n approximate integration of this equation has
been suggested in ref. 5 "by Fage and Falkner for the conditions
in which
T = T + Bx
w a and vu = Cx (4.25) -. r I "8That i s , with a power law v a r i a t i o n for the vrall temperatvire and
surface pressure. Following the method of t h i s reference, i t i s
possible t o deduce the c o r r e c t i o n i n s l i p flow for these same
conditions, on the b a s i s of the aralogy we have already formulated
h e r e . This i s done i n Appendix I I . The r e s u l t obtained i s t h a t
the changes i n the Nusselt Number or skin f r i c t i o n coefficient
( i . e . the values of A ( N U ) = NU - Nu*, or A k j j = k^ - k^*)
are givan "by ±ho rola-tion. Cui.C) ot -thiSL-b Appc^ncLix^ -sriiioh x a ^
where F is a function, shown fpraphically in fig. 2. The factor
accounts for the di.ffereno3 betvreen the temperatvire at the
apparent surface of no-slip (T^) and that of the wall (T ) . If it is zero, then T^ = T, , and there is no correction required to N u or kj^ for slip effects. In fact, it appears to be positive
(see para. 4-4). On the other hand the sign of the correction also depends on N u and F ( X ) } F ( X ) is only positive for
X ^ - 2", in which range it is shown in fig. 2. Thus we may make certain qualitative deductions from eq^jation (z».. 26).
(a) In particvilar, if the wall temperature and the velocity outside the boundary layer are constant (r = t = O ) , then it follows from (4.26) and fig. 2 that the correction djie to the temperature-jump is zero. This is an important result as it is the condition most nearly realised in practical examples.
(b) If the wall temperature is constant, but the velocity outside the boundary layer increases (t = 0, r J> O ) , the correction to Nusselt number is negative, implying a reduction in the heat transferred to or from the body. On the other hand, if the velocity decreases (r < 0 ) , the correction is positive, implying the opposite conditions.
(o) Again, if the velocity outside the boundary layer is constant (r = O ) , but the temperature decreases av/ay f rem the nose along tlie surface, the correction to the heat transfer coefficient
(i.e. /iik;,) is negative, implying an incremental heat transfer from the body to the air. If the temperature increases aft of the nose, the correction has the opposite sign, and there is either an increase in the heat transfer to the body or a daorease in that transferred to the air.
4.52. .^n Extension to .the Compressible Boundary
To deal satisfactorily with the same kind of problem in compressible flow, (i.e. vlthout introducing any assumption
as t o the maEni+rmac of M ) , i t i s necea.=!ary f i r s t t o IrapoGe the
condition t h a t the flow i s I s o b a r l c , i . e .
f =0» P5 =Pa (^-27)
so that it is no longer possible as before to ascertain theinter-action of changes in pressure and wall temperature as in inccm-pressible flow. Further, we must assume constant specific heats and Prandtl number, and follovdng the method of ref. 6 we deal with the, flow properties if the viscosity varies linearly with
temperature: i.e.
CO = 1 (4. 28)
in our previous notation. The argument is detailed in Appendix III. As a particular result of the above assumptions, it follows that the shear stress distribution is independent of the temperature distribution, so that its correction for slip is identical with that derived previously in para, 4.2. Then because of (4.2?) above,
Z^c^ = 0, and c^:=c/=0--pt (4.29)
X
m x:
If the wall temperature .is expressed as a power series
\
= ^th ^ ^a 2 B , s^ (4.30)
n *
v/here T,, is the thermometer temperature in no-slip flow, it is shcmn in Appendix III that the correction to the heat transfer coefficient is given by
Ri A ^ . - / Ï [(1^) (H-ic/)^ ^ fe) m - (¥}] i >% -^]
(4.31)
where Q) i s a f a c t o r depending upon the temperature d i s t r i b u t i o n
of equation (4.30), and i s given by
J =|"i>l^(^-^^)^(-t)Bt'
t - 1t
1I2
2F(t)B.s^-^^ (4.32)
t
t
J
Hence F is approximately the same function as is shown in fig. 2; the difference arises only because of the approximations introduced in ref. 5 (from which the values of F were calculated)which were
-24-not introduced in ref. 6 (from v^iich the above result was derived).
The first term (depending on c*) in (4.31) represents the effect of slip on the heat transfer, and the second (involving the accommodation factor a) represents the effect of the temperature-jump.
If k^ = 0, then k^ is small, and (4.3l) is in agree-ment with the equation (4.22) already derived for this condition
(if we put p = pg and 'o = 1, as given by equations (4.27) and (4.28)).
If M is small, (t* Nu*) is a quantity of order (l/!I )
c o- * 2
and so is large compared with (R^ c„ ) which is of order unity. Thus in this so called 'incompressible flow' condition, the latter
term may be neglected, and frcxn (4. 31)
f = -/F[(l^)(r^)-(¥)]-f(^D#
Noticing that for a temperature distribution of the type
T = T* + T B, s* = T (1 + B, s""^) if M = 0.
w th a t a t '
then from (4. 32)
5 = F(-G4) /F(t) •
It v/ill be seen t h a t the new r e s u l t i s quite compatible with t h a t of (4.26) previously obtained for the incompressible flc"' condition, i f p = pr (and so r = O).
•a
If the surface temperature is uniform (whatever the Mach
number), i n (4.32) v/e have t h a t t = 0, and since F ( - | - ) = 0 i t a l s o follows t h a t 0=0. Thus, for T = const, using (4.24)
J— w
E'A^j = - , / V iFf; ^^^" V =- s (~F) ••••' '^^^'
which is an iraporta.nt relation, since in most applications the temperatvire of the wall at least does not greatly vary.
If it were to vary we shovild have, in accordance with (4,31), to add an increment to A k ^ , v/hich (as in the pre-'/iously discussed condition of incompressible flow) is positive or negative, according to whether the vail temperature is (in general) either increasing or decreasing downstream. In the usual flight condition the skin temperature is highest near the nose, so that the correction
(4.33) would slightly underestimate the reducj'tion in the heat
transfer to the skin resulting from considerations of the finite molecular free path.
Because V7e can construct the correction to the heat transfer coefficient resulting with a given skin temperature distribution, it should be possible to solve the inverse problem and in particular to establish the correct value of T., , the
t h '
thermometer temperature, ( i . e . the temperatvire of the' rrall i n the
condition of zero h e a t - t r a n s f e r ) . I t would be expected t h a t
t h i s value v/ould d i f f e r only l i t t l e from i t s value T., i n
no-s l i p flo«.7. Hovrever, ano-s i no-s f u l l y dino-scuno-sno-sed i n the Appendiceno-s I I I
and r^, t h i s condition involves a s i n g u l a r i t y i n the mathematical
solution, and the known bound^iry conditions do not prcjvide us
with the information necessary to determine the precise difference
between T,, and T , . Ho-.7ever, i t i s found t h a t the ' e q u i l i b r i u m '
condition of zero heat t r a n s f e r c^n only be reached i n s l i p flcvV
a f t e r heat has been l o s t from the boundary l a y e r t o the e x t e r n a l
flcTw; or, i n other words, the thermal l a y e r then extends outside
the v e l o c i t y l a y e r . More p r e c i s e l y , whereas i n n o - s l i p flow the
temperatvire decreases exponentially as the square of the distance
frcm the surface, i n s l i p flcjw (in the condition of zero-heat
t r a n s f e r ) i t decreases merely inversely with t h i s distance. The
physical i n t e r p r e t a t i o n of t h i s r e s u l t i s obscure: i t •'.TOuld
suggest t h a t the equilibrium condition would not be reached as
quickly i n the presence of s l i p as otherwise, because an i n f i n i t e
flux of h e a t from the boundary layer t o the a i r i s necessary t o
a t t a i n t h i s condition. On the other hand, our approximations
may not be adequate for the treatment of t h i s p a r t i c u l a r condition,
the true value of T,, being dependent on conditions near the
nose v/here our theory breaks down.
4.4, The Accarmnodation & I'lcmentum Transfer Coefficients
The numerical evaluation of t h e corrections t o be
a p p l i e d t o the skin f r i c t i o n and heat t r a n s f e r c o e f f i c i e n t s
depends on a knov/ledge of the c o e f f i c i e n t s a and f.
I n s u f f i c i e n t experimental information e x i s t s to make more than
a rough approximation t o these v a l u e s .
Laboratory t e s t s suggest the follcrilng r e s u l t s f o r ' t h e
value of the accommodation c o e f f i c i e n t :
(i) a increases v/ith the molecular weight of the gas
(e.g. from 0.2 for H_ up t o 0.8 for N^ or 0^ on a platinum
surface)
-26-(ii) a increases vlth surface temperature (e.g. for H„ on Pt, from 0.2 to 0.5 as the temperature is raised to 100ü°C from rocm temperatvire)
(iii) a depends on the surface finish or quality (e.g. a rises from 0.3 - 0,4 up to 0.7 - 0.9 if a platimum surface is blaclcened with platinum black)
(iv) a depends on the age and history'- of the solid boundary (e.g. although a is recorded as 0.2 when a fresh gas-filmed surface is tested, the value a = 0.8 is obtained for an old, but otherwise similar, specimen surface),
(v) a depends upon the material of the solid surface. The following values are talcen from ref. 7 and refer to air on:
cast iron, polished a =0,87 - 0,93 machined a = 0.87 - 0.88 etched a = 0.89 - 0.96 aluminium, polished a s 0,87 - 0.95 machined a = 0.95 - 0.97
Results for the value of f, the momentvm transfer coefficient, display similar trends, and in every case where a ccmparison is possible, it is found that f is significantly larger than o,, though likewise it has never been found to exceed unity: (e.g. for hydrogen on glass a = O.36 but f = 0.89).
All this evidence suggests that for anything other than fresh, clean, cold 'laboratory specimens' the value of a is at least 0.7 and such a figure corresponds only to the results of tests using light gases. On the kind of manufactured surface which an aircraft might possess, in the presence of air possibly
supplying heat to it, we might therefore expect that a would be between 0.9 and 1.0, and that f would be virtually unity.
The value of cr is also required to evaluate the
corrections, and vre have that cr = •jS , Y = r » for air, (approx.), It occurs in combination with a and y as the expression
Y+1 \o. (Xj
which, for a i r , has the value 1.75 i f a = 0.95. We would t h e r e
-fore be safe i n assuming t h a t
r ^ /2-oA 2-f'I s. ^
and it follows that, from (3.15), in the presence of heat transfer to the wall, the value of T^ (the temperature of the air at the
plane of 'no-slip' below the true surface) exceeds the wall tem-perature, as also does the temperature of the air at the wall.
4.5. Limitations and Applicability of Solutions
In tlie preceding paragraphs we have obtained precise e:cpressions for the corrections to be applied to k!^ and c* for the ccmpressible boundarjr layer with no heat transfer; the case of the compressible boundary layer with heat transfer but without a surface pressure gradi.ent is also considered, subject to the assumptions that the Prandtl Number is unity and that the viscosity is proportional to the absolu.te temperatvire. The last mentioned assumptions are not essential, as v/e have seen, only if M = 0, but they are probably sufficiently accurate for most conditions
encountered in subsonic flow.
YIe have not been able to derive the corrections to be • applied in the condition of heat transfer at high Mach number; nor have we been able to deal with the condition of heat transfer in the presence of a surface pressure gradient, except for the condition that M = 0.
Presuaably some of these problems could be solved by introducing a^ppr-oximations, but this vlll not be attempted here. The difficuJ.ty lies in the variat3.on of the temperature-jump along the surface: the assumption of a mean constant temperatvire jump is vtnsatisfactorj'", as vre have seen that the form of the corrrection to be evaluated depends critically on the form of the variation of temperatur e - jvimp.
However, we have considered a fairly vd.de range of conditions and it is important at this stage to examine the
applicability of the solutions. The corrections we have evaluated do not enable us to determine either the total skin friction or the total heat flux to the surface. This point may best be vmderstood by considering the correction to the rate of heat transfer. It is m.ade up of t-wo terms, in general» the first depending on the effects of slip, which varies as (c„)", and the
. 0 second on the effects of the temperature-jump varying as (ki^) . The first term (if not the second, also) varies as l/x, so that although it is in general small, it is not so,near the origin of the boundary layer. In fact as x — ^ 0 , the correction exceeds the value of the uncorrected heat transfer coefficient. This is plainly a contravention of our initial assumptions that the
correction due to slip is small, and it arises because the boundary
TECHNISCHE HOGESCHOOL
VUEGTUIGBOUWKUNDE Konaaalstiaot 10 - DELFT 2 8
-l a y e r equations are no -longer v a -l i d near the o r i g i n of the -l a y e r .
An examination of the c o r r e c t i o n terms shows them to be of order
\f M / R compared vdth the unccarrected values, and evidently they
are not small (as assumed) i f R becomes a magnitude of order
2 ^
M . In such a condition the approximations involved by our
a n a l y s i s become i n v a l i d . S t r i c t l y , then, our r e s u l t s apply only
if
R»lvl2 i.e. f » | • • '
^^L J
or x > > — ~ I.e. ?->>i;.
• aThe restriction is unimportant in relation to the
values of the rate of heat flux, since usually such values are not
needed near the origin of the boundary layer, and the local rate
of heat transfer downstream of the nose is a quantity v/hich has
greater significance in boundary layer theory.
However, the local rate of shear stress is rarely a
criterion of much significjanoe: the total skin friction force is
the property v^hich is required to be knc3\»ni, and in this connection
the present analysis is of little help. Even if dcnmistream the
correction to the local shear stress is zero - as is the case, as
vre have seen, if the pressure gradient is zero « v/e may not infer
that this is also true near the nosei in fact, certainly the
shear stress must be finite near the nose, since the momentum
carried to the surface by the impinging molecules must be finite.
On the lines of the kinetic theory of gases, the number of
impinging molecules may be calculated as T N v (in the notation
of Appendix l) and each carries at most a sideways momentum mU,
where U is some velocity which will be commensurate vdth that
of the surface relative to the air. The maximum transfer of
momentum is thus
7-Nv mU = 1 p V U =
\-
^
4 4
"^ 4c
A
or in other words,
(cj =o/-AA =of^l =r » «^y.
^ f "^max \p M \ j V-^'v ^'•''Hence, if c„ is the total skin friction and c„ is its value
uncorrected for slip effects
0^ - (t <- r(c„) - c* 1 ds
f f ^ j t: f^max f J
U o
if c„ = (c„) at s = e. But in general we find that, in the
I I max -1 -\ .| '
absence of a pressure gradient, R'^c^ = (R'- C*)S'~^ where (R^ C^) has a finite constant value given by equation (4.29). Hence, vne calculate that
2
~\^ K / R *
and so
R^ (Gp- c ^ X - CR^cp^-4: (4,35)
Thus there v/111 be a first order negative correction term to C^ iny/M /R, i.e. one which is coiaaensurate with those v/e have been investigating by the present discussion, - and such a correction is due to the modification to the boundary layer flow near the nose, and not at positions dovmstream,
Practically, then, the use of the results v/e have derived here, to accovint for slip effects on the shear stress at the surface dov/nstream of tlie nose, vdll be to enable a check to be made of more approximate theories which may account for the changes near the nose as well as do-wnstream. This -«e shall there-fore attempt to do in the next paragraph,
The results for the correction to the heat transfer coefficient are, on the other hand, more likely to be of some practical value. In this connection it is worthv/hile noting
the conditions of flow likely to necessitate use of the correction. Our corrections are in general (for finite Mach numbers) of the order of ( I / R ^ ) , as already noted: and we have also noticed that there are other corrections, vrhich we have neglected, of the order of (l/R ). If then we are satisfied vdth a numerical estimate accurate to within 1 per cent, the corrections v/ould be inappreciable if the local Reynolds number exceeded about 10 ,
and inaccurate (-within the stipulated margin) if the Icxial 2
Reynolds number were belov/ 10 . On the other hand, if we were satisfied with a wider assessment (with seme 10 per cent margin of error) then it vrould be legitimate to apply the corrections dovm to Reynolds numbers of 10. In making these suggestions, it must be remembered that there is a great danger in interpreting magnitudes, which are assumed (mathematically) infinitesimal, as numbers. The presence of a numerical factor (of 0.,1 or 10, say) to be applied to such a magnitude v/ould of course make no
difference to the theoretical reasons for neglecting or including it, but nevertheless would make a great difference to its numerical
importance in calculations.
-30-5. Ccjraparison with Other Theoretical Results
The problem of slip flow in relation to the boundary layer has so far attracted little attention. In a short note, Schaaf has approached the problem of the effect of surface slip on the skin friction of an inccmpressible boundai-y layer over a semi-infinite flat plate using Raleigh's method - i.e. by con-sidering the one-dimensional growth of a laminar layer with time. The method is an elegant and simple one and yields a value of the local skin fx-iction coefficient c^, v/hich may be expressed as
•"• 2 •
the asymptotic series valid if R >j> M',
X
1 ""
Rj °f = 0.664
1 - 0.380 | - + . . .X
• • « • • # • • • « • « \_) m I J
This T/as shcjvm i n ref. 9 which contains a g e n e r a l i s a t i o n of
Schaaf's r e s u l t s t o compressible flow on the b a s i s of the
assump-t i o n s we made here i n . p a r a . 4.32. Because assump-there i s no assump-term i n
the r i g h t hand side of (5.1) i n V M / R , i t i s evident t h a t t h i s
r e s u l t i s i n agreement with the deductions we have made here,
-t h a -t -there i s no f i r s -t - o r d e r e f f e c -t of s l i p on -the l o c a l r a -t e of
sldLn f r i c t i o n . The term i n (M / R ) we would not find by the
method outlined i n the present report as terms of such order are
the e r r o r terms i n the o r i g i n a l boundary l a y e r equations. Having
i n mind the f a c t that no account i s taken of the non-linear terms
in v i s c o s i t y i n the equations of r e f . 8 as well, i t i s open t o
doubt whether the resultsof t h i s reference have any g r e a t e r s i g n i f
-icance than those of t h i s r e p o r t . Hovrever, as i s suggested i n
ref. 9, the r e s u l t s might be a guide to the conditions e x i s t i n g
even i f the assumptions are inaccurate, and i t i s noteworthy
t h a t the ansvrer obtained f o r the val-ue of c„ i n free-molecule
flow (where R « M ) i s only tv/ice t h a t developed more accurately
by Ashley i n ref. 10, Hence the r e s u l t s do not appear to be too
iriaccurate and may be applied t o c a l c u l a t e the flow near the
o r i g i n of the boundary l a y e r so as t o give the value of
1
R2
0^ = 1.328
1 - 0 . 7 7 3 ^ + 0.380 | - + . . . ( 5 . 2 ) M „ , 0 ^ M^ R2 ^ JAs noticed in para. 4.5, a first-order term in (li^/fe-) arises due to the behaviour of the flov/ near x = 0. No doubt inclusion
/of other ,
1 2
i A recjent discussion ' of the effects of slip on the skin friction on a flat plate in incompressible flow has included the particular results of the present work: (a) that A c^ = 0 to a first approximation, and (b) that there is a reduction in
of other terms in the equations of motion might enable a more accurate answer to be obtained; but it must be admitted as un-likely that any single solution on such lines could accurately describe at the same time both types of free-molecule and contin-uum flow. An approximate treatment must be undertaken, and Schaaf's results seem adequate.
The expression of (R'^C^) in terms of powers of (^S/R")
in equation (5.l), bears out the assertion, made originally in ref. 4, that it is this parameter which is the important criterion of 'superaerodynamic' phenomena. This is also borne out by the
present results, as will be seen from an examination of the i _i_
formulae for R-c^ and R-k^ inequations (4.18), (4.22), (4.26) and (4.31). However, it •s^lll be observed that the coefficients of the pov/ers of V M " / R in these equations, whilst in general finite for M = 0, do in fact vary vlth Mach number, so that for large values of M it may be some other parameter involving M and R which is important. Thus, as shovm in ref. 3, the
2
significant parameter for slip effects if M » 1 , is in fact, V'ïï/fe, However, this is not a very important difference.
The third-order Boltzmann Equation has been used by Schamberg to establish the effects of the inhaaogeneity of the
i
air on the Couette flow, and by the inclusion of the non-linear terms in viscosity and heat conduction, a much more accurate treatment may be made than that attempted here. On the basis of these results - vdiich show the v.^ll-established decrease in shear stress v/ith slipping, - Schamberg suggests that there is also a decrease in skin friction of a similar magnitude due to the slipping of the flow in the boundary layer. On this basis he suggests, in fact, that
c 2
—
» 1 - 0.81
A - +
(0.66 + 0.10
M^) I-
(5.3)
c„ R- X f X
v/hich indicates a first-order reduction in c- due to slip. This result follov/s frcm Schamberg's iTork, howetrer, becjause he assvsnes that the boundary layer thickness is unchanged: we have suggested that the boundary layer thickness is actually reduced by slipping.
To emphasise this point v/e note that in a Couette
flcïw slipping at either of the two plates must produce a decrease in the velocity gradient in the flow betv/een them, and so also a reduction in skin friction. However, in a boundary layer f lew there is no second surface apart from the wall: if we must imagine it to have a finite thickness (a concept v/e have not elsewhere envisaged) then the results of this note suggest that,
^ /although .,.
-32-although the air slips at the surface, the boundary layer thickness is reduced (by a distance v/e have termed ^ ) , so that there is not necessarily a decrease in the velocity gradient across the boundary layer. (indeed, under certain circumstances vre have seen that there may even be an increase in skin friction).
Furthermore, to draw an analogy between Couette flow and boundary layer flow, would also imply that there is slipping of the air at the outside of the boundary layer which is plainly not so since there is no shear between the ' outside' of the boundary layer and the external flow.
These points are important, not becavise in aiy way they detract from Schamberg's remarkable paper, which has as its main theme a different problem from the one discussed here, but because it illustrates what is apparently a mistalcen analogy, which one is tempted to use in describing slip effects in a bovmdary layer,
The effect of slip flow on heat transfer has been 11
considered by Dralce and Kane, and here although there is no precise agreement with the present results, a decrease is predicted in the rate of heat transfer due to the temperature
jump: hovrever, as the results when applied to the continuum p
regime (M / R < ' < I ) yield a value for the rate of heat transfer which is four times as large as that derived frcm the usual boundary layer theory, a numerical compariscaa v/ith the results
of the present v/ork could hardly be expected to provide more than qualitative agreement. Like Schaaf's v/ork on the shear stress, this work sets out to include the full range of flow conditions
-fron continuvim t o free-molecule flow.
6, C onclus iona
(i) V.''e have developed in this report a method of finding
the first-order effects of the existence of a small but finite mean free-path csf the air molecules on the local rate of shear stress and heat transfer.at a surface in the presence of a laminar
boundaiy layer.
(ii) The results are applicable to any type of tiTO-diraensional surface, but it should be noted that if the surface is curved, error terms arise in the expression cf the values of c^ and k^ due to the surface curvature which are generally neglected in the usual treatment of the boundary layer theory, but which are in fact commensurate with the other corrections we derive here.
(iii) The method adopted is to express c_ and ki. as a
pov/er series in the parameter ( I / R ^ ) . The first term in this
series is the value usually found from boundary layer theory assuming the air to be a continuous medium; the second term in the series is (if the surface is plane) the first-order effect due to surface slip and temperature-jump, and is the tern v/e derive here. The third and higher terms denote the error in our treatment: such terns can only accurately be derived if Vv^e relax the approxl!':iations used by Prandtl to describe the equations of a thin boundary layer, and if vre include the non-linear terms in viscosity and conductivity in the statement of the equations of momentvm and energy,
(iv) Due to the singular beliaviour of the properties of the boundary layer near its origin the assumptions implicit i.n the construction of Prandtl's boundary layer equations (and in the present analysis) do not apply at positions on the surface where the local Reynolds n^ijimber is no longer high. Thus our results are applicable only as corrections to the shear stress and heat tï-ansfer downstream of the nose. This means that v/e cannot estimate the total correction to the skin friction drag, and a qualitative examination of the flow conditions near the c^^igin of the leading edge shows tliat, in this region, thei-e is a
correction to be applied to the total intensity of skin friction (as usually derived frcm boundary layer theory) which is
cocmensurate with the total correction due to slip dcjwnstream of the nose»
(v) In the practical applications of the knowledge oi the heat transfer from a boundary layer, it is usually tlie local ~ as distinct fron the tota.1 - rate of heat flux which must be
evaluated, and the results v/e derive here are of use in estimating this Icxjal value except at positions vyithin a distance frcm t]ie nose coomensurate v/ith the molecular mean free path (see the expressions (4.34)).
(vi) It is shown that, within the accuracy of our solution, the expressions derived frcm the kinetic theory of gases for the slip velocity of a gas (in a, simple shearing motion) at a surface, and the temperatvire-jump at a surface exposed to stationary gas
(through which heat is being conducted), are valid first-approxi-mations to the slip velocity and temperature-jump at a surface within a boundary layer.
(vii) The values of the slip velocity and temperatvire-jump are expressed in terns of tv/o coefficients (the coefficient of momentum transfer, f, and the accommodation coefficient, a) whose precise value must be obtained from experiment. Some further rena.rks on their values are given in para 4.5.
