A Note on the Algrabraic Representation of The Preston Tube Calibration

Cranfield

March 1983

H

The Preston Tube Calibration

by D.I.A. POLL

TECHNISCHE HOGESCHOOL DELFT LUCHTVAART- EN RUIMTEVAARTTECHNIEK

BIBLIOTHEEK KIuyverweg 1 - DELFT Aerodynamics Division

College of Aeronautics Ccanfïeld Institute of Technology

Cranfield

March 1983

A Note on the Algrabraic Representation of The Preston Tube Calibration

by D.I.A. POLL

Aerodynamics Division College of Aeronautics Cranfield Institute of Technology

Cranfield, Bedford, U.K.

ISBN 0 902937 83 9

£7.50

"The views expressed herein are those of the authors alone and do not necessarily represent those of the Institute."

A Note on the Algebraic Representation of the Preston tube c a l i b r a t i o n .

by

Dr. D.I.A. Poll

Sumfiia r y .

Algebraic expressions are presented f o r the f u l l range of the Preston tube c a l i b r a t i o n . Unlike the o r i g i n a l formulae presented by Patel the present expressions are continuous throughout the range of v a l i d i t y w i t h a maximum d e v i a t i o n of 1.6% from P a t e l ' s r e s u l t s . I n a d d i t i o n the variables are cast i n natural logarithmic form to f a c i l i t a t e t h e i r use on those mini computers which recognise the BASIC language.

List of Symbols,

d Preston tube external diameter

Ap pressure difference between Preston tube and undisturbed

wall static.

U velocity

U^ Friction velocity ( ' ^ / p )

X In i^i*^ /^e"^^^

y In ( '^^/if^v'-)

Z distance from the solid surface

V kinematic viscosity

p

density

1

-Introduction.

The Preston tube is one of the simplest and most convenient methods available for the measurement of local turbulent skin friction. It consists of a Pitot tube which is fixed to the surface with the open mouth located at the point where the wall shear is to be measured whilst the axis of the tube is aligned with the direction of the local shear stress vector. The operating principle relies on the empirical observation that for turbulent flows with modest pressure gradients the "Law of the Wall" is universal i.e. there is always an identifiable region in which the velocity profile has the form

-U = A log (Ut. l\ + B U ^ V V /

when A and B are constants. I t follows t h a t i f the diameter of the Preston tube i s s u f f i c i e n t l y small f o r the tube to be completely

submerged i n the "Law of the Wall" region then a universal c a l i b r a t i o n of the form

e x i s t s .

Although several attempts have been made to determine the calibration it is generally agreed that the results of Patel are the most accurate. To facilitate the use of his data Patel provided a series of algebraic expressions which covered the complete range of his experimental results. However it has been

2

pointed out by Head that there are discontinuities in the calibration if Patel's expressions are taken literally. It is the object of the present note to augment Patels original formulae so that these

discontinuities are removed and also to provide modified expressions and algorithms which are more suitable for use on mini-computers.

2

-The Algebraic Expressions.

In Patel's notation the dependent and independent variables were

* *

y and x respectively where

However it is common practice for current generations of mini-computers to operate with the BASIC language which does not have a log,« function available as standard. Therefore in the present context the dependent and independent variables will be

Therefore

y = 2.302585y* x = 2.302585 x*

a) For X 4 6.5862 (x = 2.8604)

y = 0.5x + 0.0852 (1) (y = 0.5x + 0.0370)

In Patel's original work, equation 1 was intended to be used for *

values of x less than 2.9 (x < 6.7). However figure 1 shows that at this point Patel's expressions exhibit a discontinuity and to avoid this a patch has been applied between x equal to 6.58617 and x equal to 6.71537. Hence

For 6.5862 < x. è 6.7154

y = 0.57387 x - 0.4013 (2) ( y* = 0.57387 x* - 0.1743)

3

-b) For 6.7154 < x ^ 11.5129

y = 1.9082 - 0.13810 x + 0.062408x^ -0.0011317x^ (3) ( y* = 0.8287 -0.1381 x* -t- 0.1437x*^ - O.OOöOx*"^)

* Patel suggested that equation 3 should be used for values of x

up to 5.6 (x < 12.89) but once again this would introduce a discontinuity - see figure 2. Since the mismatch between the functions is of order 3% rather more care has been taken in the choice of the 'patching' polynomial. In this case a f i f t h order polynomial has been used which matches Patel's expressions f o r displacement, gradient and curvature at an x*of 5.0

(x = 11.51) and 6.0 (x = 13.82) i . e . for 11.5129 < x ^ 13.8155 y = -1231.3564 + 495.50939x - 79.405663x^+ 6.3516497x^- 0.25317676x^ + 0.004023461x^ (4) (y* = -534.7713 + 495.50939x* - 182.838297x*^+ 33.6757994*^ - 3.09079994x*^ + 0.113099998x*^)

N.B. Derivatives of Patels expressions are given i n Appendix A.

c) For 13.8155 < x ^ 18.3

X = y + 2 In (0.8469y + 4.10) (5) (x * = y* + 2 log,^ (1.95y* + 4.10) )

4

-A Solution Method For Equation (5)

I t w i l l be noted that equation (5) is i m p l i c i t i n y and therefore must be solved by an i t e r a t i v e technique for known x. I f the equation is re-written as

y = X -2 In (0.846874y + 4.10)

where x is now a constant then a solution may be found by treating this single equation as a pair of equations which must be solved simultaneously i .e.

Fi (y) = y (6) and F^ (y) = x -2 In (0.846874y + 4.10) (7)

Since the range of v a l i d i t y of equation (5) is such that y l i e s between 8.9 and 12.9 i t is clear that equation (7) exhibits very weak dependence upon y. Therefore i t is proposed that following an estimate of y , say yj^, equation (7) should be approximated by a straight line passing through and tangential to the function i . e .

^2 (y) 1Ü d<'2\ • (y - yf^) + ^i (y^)

Wly = y^

F2 (y) « f - 1.693748 "I ( y - y j + x - 2 In (0.846874y^,+ 4.10)

L(0.846874yfj + 4.10)J ^ ^ (8)

A improved approximation to the solution of (5) is then obtained by solving equations (6) and (8) simultaneously i . e .

1.693748y^ + (0.846874y^, + 4.10)(x -2 In (0.846874y^ + 4.10))

y(N+i) = ^ Ï! ^

(0.846874yM + 5.793748)

^ (9)

1.693748y*fj + (1.95y*,^ + 4.10)(x* -0.8685891n(l .95y*^ + 4.10) )

^*(N+1) =

(1.95y* + 5.793748)

It is found that if y© is taken to be 8.9 a solution correct to better than 6 decimal places is obtained after three applications of equation (9).

5

-References.

1) Patel, V.C. Calibration of the Preston tube and limitations on its use in pressure gradients.

Journal of Fluid Mechanics, Vol.23, Part 1, p.p. 185 - 208, 1965.

2) Head, M.R. Simplified presentation of Preston tube calibration. Vasanta Ram,V.The Aeronautical Quarterly, Vol. XXI1, pp 296 - 300,

Appendix A. Derivatives of Patel's Expressions y* = 0.8287 - 0.1381 x * + 0.1437x* - 0.0060x*' dy* = - 0.1381 + 0.2874X* - 0.0180x* ÏÏÏÏ* d^y* = + 0.2874 - 0.0360x* dx *2 * * y = X dy* = 1 -• -• * - 2 In(lO) 2 ln(1.95y* + 4.10) ^ X * X 1.95 dy dx In(lO) (1.95y" + 4.10) ' " ^ x ^ or dy = ?F dx 9 * 1.95y* + 4.10 1.95y + 5.7937 3.3028 (1.95y* + 5.7937) (a) (b)