CoA Report A e r o No. 173
BlEUOTHfctK
THE COLLEGE OF A E R O N A U T I C S
C R A N F I E L D
THE USE O F PRESTON TUBES TO MEASURE THE SKIN FRICTION IN
TURBULENT BOUNDARY LAYERS WITH SUCTION OR INJECTION
by
CoA Report Aero 173 May, 1964
THE COLLEGE OF AERONAUTICS
CRANFIELD
The use of Preston tubes to measure the skin friction in turbulent boundary layers with suction or injection
b y
-T. N. Stevenson
SUMMARY
A theory is presented which shows how Preston tubes may be used to measure the local skin friction in incompressible turbulent boundary l a y e r s over a porous surface through which there is a small suction or injection velocity. The skin friction measurements obtained in an experiment using Preston tubes a r e in good agreement with those obtained using the integral momentum equation,
Contents List of symbols 1. Introduction 2. Theory 3 . Experiment 4. Conclusions References Tables Figures Page No.
L i s t of s y m b o l s
2a i n s i d e d i a m e t e r of P r e s t o n tube 2b o u t s i d e d i a m e t e r of P r e s t o n tube B c o n s t a n t in the law of the wall equation c - l o c a l skin f r i c t i o n , r—^ f P u 2 1 C c o n s t a n t in equ. 4 . d = 2b, o u t s i d e d i a m e t e r of P r e s t o n tube f function of —— only ^1' ^2' ^> ^4 functions of t only K von K a r m a n ' s c o n s t a n t n c o n s t a n t in equation 4 .
( P - p ) p r e s s u r e r e c o r d e d by the P r e s t o n tube r e l a t i v e to the s t a t i c p r e s s u r e .
u v e l o c i t y in the x - d i r e c t i o n Uj^ f r e e s t r e a m v e l o c i t y
f r i c t i o n v e l o c i t y , / - 2 = u^ / ^
" T
V ^ v e l o c i t y n o r m a l to the p o r o u s s u r f a c e at t h e s u r f a c e
X c o - o r d i n a t e along the c y l i n d e r in the f r e e s t r e a m d i r e c t i o n (the o r i g i n i s the beginning of the p o r o u s section)
y c o - o r d i n a t e n o r m a l t o t h e s u r f a c e of the c y l i n d e r # y - b = a Sin # V k i n e m a t i c v i s c o s i t y p d e n s i t y o a r e a of P r e s t o n tube opening •^w skin f r i c t i o n , {= ^i -§—) ^
1
-1. Introduction
It is very difficult to obtain an accurate measurement of the local skin friction in turbulent boundary l a y e r s . Pitot tube t r a v e r s e s may be used together with von Karman's momentum integral equation to relate the local shear s t r e s s to the changes in the momentum thickness, but the method r e q u i r e s the differentiation of experimental r e s u l t s in the streamwise direction which is rather inaccurate, and the method is also very sensitive to three dimensional effects.
Accurate measurements of the velocity profile very close to the wall have been attempted in order to find the velocity gradient and hence the skin friction, but large corrections to the instrumentation calibrations a r e required due to the presence of the wall, and the method is extremely difficult. Successful measurements ^ • ^have been made with skin friction balances which consist of an isolated portion of the
surface connected to strain gauge balances. Ludwieg"^'^measured heat transfer r a t e s to the wall and related these to the skin friction and it was shown that the 'law of the wall' held in p r e s s u r e gradients just a s it did in zero p r e s s u r e gradient.
Stanton tubes and Preston tubes may be used to estimate the skin friction. Stanton tubes^are very small pitot tubes which a r e used in the linear velocity profile in the sublayer region and Preston tubes a r e pitot tubes which a r e used in the universal logarithmic regions. Preston tubes were originally calibrated in pipe flow on the assumption that the law of the wall was the same in pipe and boundary layer flows, but there is now some doubt a s to the exact calibration curve (see the discussion in references 7 and 8). Further work is continuing to determine the best calibration curve since the use of Preston tubes is the easiest method of estimating skin friction.
The theory presented in this paper shows how Preston tubes may be used to estimate the skin friction in turbulent boundary layers over porous walls through which there is a small normal velocity. The theory gives an equation which may be used
with the Preston tube calibration curve (whichever calibration curve is eventually chosen). The theory follows that of Hsu^but the equations now include the suction or injection velocity at the wall. The final equation is relatively simply to apply and the skin friction r e s u l t s which a r e obtained in an experiment compare favourably with those obtained using the integral momentum equation.
2. Theory
The 'law of the wall' equation for turbulent boundary layers with zero p r e s s u r e gradient and zero transpiration velocity, which is valid in the overlap region, is
* u 1 y^T
The 'law of the wall' equation is — = ^ logg - ^ + B, where K and B a r e constants, u is the velocity in the inner region of the boundary layer at a distance y from the wall and UT is the friction velocity ( = / — i
••"The overlap region is the region in which both ttie inner and outer solutions a r e valid. The inner solution is one of the form — = f l-rr- J and the outer solution is of the form
u.p \ V /
2
-g
where K and B a r e constants. Hsu fitted a power law profile to this region of the form
where C and n a r e the constants which best fit the experimental results. (Hsu used the values, C = 8.61 and n = 1 ) .
T
If there is suction or injection through a porous wall then the law of the wall equation is modified to
i 2 i i - r / V ^ UN
Zurf /, ^ w " \ * , •) 1 , yur (3)
This equation was presented in reference 10 where it was shown by comparison with experimental r e s u l t s , that B and K a r e independent of V^ and u.^,and therefore take the values for the case of zero transpiration. This is also shown by a dimensional analysis in reference 11, Some experimental results from reference 12 are plotted
2u . V a s ^
Vw
I . / M + _ï!—\ - l \ against log —^ in figure 1 and a r e compared with an equation of the form
2u / / v... u\ 2
The equation shows reasonable agreement in the overlap or logarithmic region and it is now rearranged to give the equation for the velocity, u;
It will be assumed that the presence of the pitot tube does not affect the flow in the boundary layer and that the p r e s s u r e recorded by the pitot tube is an average of the integrated p r e s s u r e over the open portion of the tube
0
where ( P - P Q ) is the p r e s s u r e recorded by the tube relative to the static p r e s s u r e , PQ,
and a r e f e r s to the area of the tube opening. A pitot tube which touches the wall and has a circular cross section of inside diameter 2a, and outside diameter 2b is considered. Equation (6) is written
, ) n a ^ = Uj 2 u V a 2 - ( y - b ) '
( P - P Q ) na^ = i p / 2 u V a 2 - ( y - b ) 2 dy (7)
or « 2
(P-pQ)»a* = ip I 2 u a . a" C o s ^ ^ d^ (8) 5.
" 2
where y - b s a Sin ^ o^^ "u = (1 + t Sin • ) ; t = — .
T h e equation for u, equation (5), i s s u b s t i t u t e d into equation (8) and the s u b s e q u e n t equation i n t e g r a t e d t o give 2n
C a u / ( ^ ) I^(t)] (9)
t / 2 w h e r e I ^ ( t ) « T (1 + t Sin«, Z"^"^^^" C o s ^^ dit> (10) m = l , 2 , 3 - | E q u a t i o n (9) c a n b e w r i t t e n , / n \ . , J S ^ a 2+4n 2+3n ( P - p ^ ) « d a f r^ * / V . . . \ ^ / ü ^ \ r^^ 4pva 3 2+2n w h e r e d = 2 b . When V^, = 0 t h i s e q u a t i o n r e d u c e s t o 1 T d* / ( P - P K 2 \ n+ï" 4pv2 o r \ 4pv2 ; "* = k( , , . " " ) (12) t d ^ P - p ) d \ w h e r e 14
-g
Hsu evaluated I, (t) for different values of t (see table 1) and showed that the value of k changes very little with t providing t is l e s s than about 0 . 5 , i . e . if a thick walled tube is used then I^ (t) - I (0). There is doubt a s to the appro-priate values for the constants K and B in the law of the wall equation and therefore corresponding doubt with regard to the values for C and n. Hsu used the values, C = 8.61 and n = 1/7, and therefore equation (13) reduces to
a = 7.6274 + 0.875P (15) when t = 0 and
a = 7.6298 + 0.875P (16)
when t = 0 . 5 , where
The calibration formula which Preston obtained for pipe flow is a = 2.604 + 0.875 P , (17)
7 the formula suggested by the N. P , L. is
a = 2.647 + 0.875 p (18) 2
arid that suggested by Smith and Walker i s
a = 2.634 + 0.877 p . (19)
If it is assumed that Ij^(t) = 1^(0) then equation (13) reduces to equation (18) •when n = 1/7 and C = 8.4 and to equation (19) when n = 0.14 and C = 8.48, and the
corresponding equations for — (from equation (2)) a r e U T 8 . 4 ( ^ ) (20) u U T and u UT 0.14 8.48 ( ^ ) . (21)
These a r e compared with some law of the wall equations in figure 2. It is difficult to decide which a r e the correct values for C and n, but this theory gives an equation of the right form and the values of C and n may be adjusted to suit the calibration curve which i s eventually chosen.
5
-When there i s a normal velocity at the wall, equation (11) must be used. C and n a r e independent of V^ and therefore have the values obtained for the
case of zero transpiration, and the integrals, I (t), I (t) and I (t), may be evaluated for the particular pitot tube which is being used in the.experiment.
Some values for I , I and I a r e given in table 2.
Equation (11) is of the form
4n 3n 2n
(P-p^) C^ = V^ UT C^ + V^ ur C^ + u,. . C^ (22)
where Cj^ , C ^ , C^ and C ^ a r e known. Curves of (P-p ) against c.f = 2(—) j may be plotted for particular values of V^ and u .
If it is assumed that I (t) - I (t) - I (t) - I (0) = ^ ,
* S 1 l a
equation (11) reduces to
3. Experiment
The model is a porous cylinder of 4 " diameter with a solid elliptifc hose (fig. 3) and is described in detail in reference 12. A Preston tube was placed ott the surface at different positions along the cylinder and the p r e s s u r e s which were measured for different blowing velocities a r e shown in fig. 4.
There is a certain length at the beginning of the porous surface during which the boundary layer is adjusting itself to the new conditions. This relaxation length Increases slightly a s the blowing velocity i n c r e a s e s . The equation for the inner region (eqn. (3)) is not valid during this relaxation period and therefore the Preston tube r e s u l t s a r e only compared with momentum t r a v e r s e s over the latter portion of the porous cylinder.
The Preston tube has an outside diameter of 0.064 inches with t equal to 0.68 and was always within the overlap region during the experiments. The free stream velocity was 50 feet per second. Curves of (P-p^) against Cf (fig. 5) were evaluated for several blowing velocity ratios using equation (23), and the curves a r e used to estimate the skin friction over the porous surface. The skin friction r e s u l t s a r e shown in fig. 6.
In reference 10 it is shown how the skin friction may be obtained for a paHiculat^ suction or injection velocity from a velocity profile using the law of the wall ecjusttlon with suction or injection (equation (3)). The skin friction results estimated in this way and those obtained using the monaentum integral method a r e compared with the skin friction r e s u l t s using Preston tubes in fig. 6, and the r e s u l t s agree very well. (The momentum integral r e s u l t s which a r e only accurate to * 10% a r e discussed in reference 12).
6
-4. Conclusions
An equation has been derived which enables Preston tubes to be used in turbulent boundary layers over a porous surface through which there is a small suction or injection velocity. The skin friction measurements using the Preston tube agree with those which were obtained by the other methods,
A cknow ledgm ent s
The author wishes to thank Professor G.M. Lilley of the Department of Aerodynamics, The College of Aeronautics, and Professor A . D . Young of Queen Mary College, The University of London, for many helpful discussions during the course of this work.
References
1. Dhawan, S. Direct measurements of skin friction.
NACA TN.2567, 1952. 2. Smith, D . W . , and
Walker, J . H .
Skin-friction measurements in incompressible flow.
NACA TN.4231, 1958.
3, Ludwieg, H. Instrument for measuring wall shearing s t r e s s
of turbulent boundary l a y e r s . NACA TM. 1284, 1950. 4 . Ludwieg, H . , and
Tillmann, W.
Investigation of wall shearing s t r e s s in turbulent boundary l a y e r s .
NACA TM.1285, 1950. 5. Stanton, T . E . ,
Marshall, D . , and Bryant, C.N.
On the conditions at the boundary of a fluid in turbulent motion.
P r o c . Roy. Soc. (A), Vol. 97, p. 413, 1920. 6» P r e s t o n , J . H . Determination of turbulent skin friction by means
of pitot-tubes,
J . R . A e . S , 58. p.l09, F e b . 1954. 7. Staff of Aerodynamics
Division, N . P . L .
On the measurement of local surface friction on a flat plate by means of Preston tubes.
R. & M. No. 3185, 1961. 8. Head, M . R . , and
Rechenberg, I.
The Preston tube as a means of measuring skin friction.
A . R . C . 23.459. F . M . 3 1 5 3 . 1952.
9. Hsu, E . Y . Measurement of local turbulent skin friction by
means of surface pitot tubes.
David W. Taylor Model Basin. Rpt. 957. NS715-102. 1955.
7
-10. Stevenson, T . N . A law of the wall for turbulent boundary layers with suction or injection.
College of Aeronautics Rpt. 166, 1963. 1 1 . Stevenson, T . N . A note on the inner region of turbulent
boundary l a y e r s , (To be published),
12. Stevenson, T . N . Experiments on injection into an incompressible turbulent boundary layer.
Values for the integral: I (t) Table 1 t 0 0 . 5 1 . 0 I i 1.5706 1.5603 1.5161 k 0.0424 0,0426 0 . 0 4 3 7 log k 2 . 6 2 7 4 2 . 6 2 9 8 2 . 6 4 0 5 Table 2 t 0 0 . 6 8 1.0 I i 1.57 1.54 1.52 l a 1.57 1.53 1.51 I 3 1.57 1.53 1.50
I
" / 2 m+1 P (1 + t Sin«) "^ . Cos2» d* ~ 2+
'M
3 0 I 7 S IS1
V EOU eou 41 1
1 WHEN K . O ' 4 I 1 » . > ' • WHEN C.C 61 t II - ' / r / /j
/ /Uf'
^r
A / 4- i " y z 7 EXPERIMENTAL RESULTS ( R E F I J ) v« U L j U T U T X o l a t 4- 0 0 1 6 ] i ' 6 D 0 0 6 I J O o i l 37-4 V 0'20 4« 4 & 0 4 S 77.« L o o , - r-ï. FIGJ.THE OVERLAP R E G I O N .^ T T
- Jj. . 5 . 0 LOG|o 1 U I + 7 I S ( R E F J' - EOU :F I G 2 . FORMULAE FOR THE OVERLAP R E G I O N .
COLLEGE OF AERONAUTICS I'» 3' WIND TUNNEL
f o a
FIG.4. THE PRESSURE RECORDED WITH THE PRESTON TUBE
FIG.5. EQUATION 23 WHEN U,«50 F T / S E C .
X INCHES
FIG.6 VARIATION OF SKIN FRICTION ALONG THE M O D E L .
