Measurement of Mean Velocities and Turbulence Structure Using Hot Wires or Films in Two-Dimensional Flow. Air Flow over a Wavy Surface Abraham E. Dukler’ and Wuu N. Chen Chemical Engineering Department, University of Houston, Houston, Texas 77004
It is shown that the usual techniques of sum and differencing of anemometer signals to calculate turbulence properties may not be valid when the flow is two-dimensional. Response equations are developed which account for the presence of the second velocity. A procedure is detailed for calculating the amplitude and direction of the mean flow as well as the statistics of the turbulent fluctuations and the turbulent shear stress. The method is applied to measurements of flow over a wavy boundary. Flow normal to the surface caused by the waves is shown to exist well out into the mainstream. The mechanism for large radial transport of momentum with the resulting large pressure drop is thus understood.
Introduction Champagne and Sleicher (1967a,b) have demonstrated that heat transfer from an inclined hot cylinder is sensitive to the tangential component of velocity along the cylinder. Their careful experimental measurements demonstrated that the “effective cooling velocity, We,” can be calculated from the following equation Wez = W12 (cos‘ /3 + k2 sin’ p) ( 1) WI is the magnitude of the instantaneous velocity vector, p
is the angle between the vector direction and the normal to the cylinder, and k depends primarily on the length to diameter ratio. Methods were given for estimating k . For very long, fine wires k approaches zero and eq 1 reduces to the cosine law previously widely used in interpreting measurements from hot wires or hot films. However, for wires purchased commercially and in common use the value of k was shown to be about 0.2. These investigators executed a carefully reanalysis of the instantaneous and time averaged equations for turbulent flow over heated cylinders and developed definitive response equations for use with hot wire or film anemometers. In an ideal X array of wires having k = 0.2 with the mean flow direction a t 45’ to each wire this tangential cooling effect was shown to cause the true mean square intensity in the direction normal to the mean flow, (?/U2), to be 17% higher and the true shear stress, (E/ U 2 ) ,to be 8% higher than that calculated from the anemometer output voltages by assuming the cosine law valid. Commercial hot films with smaller length to diameter ratios show a typical value of k of 0.5 and then the error in u2/U2is 28% and in E / U 2 is 67%. No error in 2 / U 2results from neglecting the tangential cooling. In using the Sleicher relationships (or the simpler cosine law equations) the angle between the vector of the mean velocity and the wire must be known a priori. For flow in smooth channels of uniform cross-sectional area the mean flow is axial and thus its direction is known, but in many experimental situations of interest this direction is not known. Particular cases of importance that have received attention recently include flow over wavy solid or liquid surfaces, flow over roughnesses, flow in entry or developing regions, and other situations of two-dimensional flow. Because the tangential component has now been shown to be important to the interpretation of the anemometer output voltages it can be expected that the component of these secondary flows which are tangent to the heated cylinder
will make further contributions to the heat transfer and must also be incorporated into the response equations. When the secondary flow varies strongly with position away from the boundary (such as in flow around roughnesses or waves) their effect change with position and the calculated distribution of the turbulence quantities can be significantly in error unless the equations incorporate these effects. In this paper the response equations are redeveloped including the influence of the secondary flow. Equations for extracting the magnitude of the mean primary and secondary flow velocity from the anemometer output are given along with equations for calculating the turbulent intensities and shear stress in the presence of this secondary flow. In principle, the Sleicher equation can be used without modification in the presence of secondary flow if two identical wires or films are used and if the array is mounted on a traversing mechanism which permits the wires to be rotated about an axis through the intersection of the two wires. The wires could then be rotated until the two mean responses are equal and this new angle remotely detected. For each position of a traverse this could be repeated. Aside from the fact that identical sensors are difficult to obtain, the mechanical contrivance is usually too complex and can introduce disturbances in the flow even for measurement in large scale equipment. It is much too large to be suitable for measurement near interfaces or boundaries. McCroskey and Durbin (1971) proposed a vee-type probe which permits measurement of the direction of the flow when it lies in the plane of the V. Very high precision is required to interpret the results and this makes it necessary to develop special high precision control circuitry not usually available commercially. The equations developed here overcome the mechanical difficulties since the wire can be kept a t a fixed angle relative to fixed coordinates and bridge circuits in commercially available anemometers can be used to obtain the signals from the wires. Instantaneous Time Average Effective Cooling Velocity Consider inclined wire no. 1 in the x-y plane as shown in Figure 1 whose normal is a t a known angle a1 to the z coordinate. The instantaneous velocity vector, W I , makes an unknown and time varying angle, p1, with the normal to the fixed wire. The mean flow normal to the x-y plane is zero. Turbulent fluctuations in this direction contribute negligiInd. Eng. Chem., Fundam., Vol. 14. No. 4, 1975 359
B, = 2(k2 - 1) tan a2
(12) Expand the right side of eq 5 and 9 by the binomial theorem, express the left side as the sum of a time average, We, and fluctuating component, we, take the time average of the resulting equation, and subtract the time average values from the instantaneous equation, all in the usual manner to obtain
we,= us1"2 cos W,, =
Figure 1. Instantaneous velocity vector and the X array. bly to the heat transfer (Champagne and Sleicher, 1967b) except for unusually high turbulence levels. Instantaneous components of velocity are designated U u in the x direction and V u in the y direction. The magnitude of the velocity vector, at any instant is
(13)
COS CY,
(14)
-+
+
w,= [ ( E +
+ (V +
U)2
(2)
2:)2]1/2
Define
(3) A series of transformations using the cosine law permits sin and cos 01 to be related to locities as follows.
01
sin2 p1
ai
a1
and the instantaneous ve-
[ ( I + Y,)~sin2 cy, + ( R + Y,), cos2 cy, 2(1 + r , ) ( R + yY) s i n cy, cos al] = (1 + Y,)' + ( R + Yy)'
In arriving a t eq 13-16 the binomial expansion was limited to terms of second order in the fluctuating quantities; terms of third order were neglected in the final equation when additive to term of second order.
Response Equations Consider a linearized hot sensor system. The voltage outputs are related to the effective cooling velocities by E(t) = E + e ( t ) ;E = KW,; e = Kw,
As a result the response equations of the system become El = K,
cos a,
(17)
(4)
Thus, substituting (4) and (2) into eq 1 gives, after much manipulation, the following expression for effective cooling velocity, Wel, for wire no. 1 wel
s,
= (1
=
u cos
+ k2 tan2
CYl)
cyl(S,
+ s, +
+ 2R(1
S,)'/,
- k2) tan
cyi
( 5)
+
R 2 ( k 2+ tan2 cyl) S2 =
It is now possible to obtain values for U and R directly
(6)
2A,R, + 2A2yy
A , = (1 + k' tan'
cy,)
A, = ( 1 - k2) tan CY^
-
R(h2- 1) tan CY^
+
R(k2
+
tan2@,)
(7)
+ A 4 r Y 2+ A5yXyy A, = 1 + k2 tan2 cyl A , = k 2 + tan' CY,
S, = A,Y,'
A : = 2(1
-
k 2 ) tan
+
(z)
R2(1
+
( 8)
B,
B, = 1
+
+
tan2
k 2 tan
cy2
k2 tan2
+
+
k2)(1 + R2) + 2(1 - k2)R kz)(l + R z ) - 2(1 - k2)R
which produces a quadratic equation for R , in terms of and 12.
Q
+
tan2 CY,)
B3 = k2
(1 (1
( ~ 2
CY,)
+ R ( k 2 - 1) tan CY, = (k2 - 1) tan + R ( l + k2 tan cy2) T 3 = B3rX2+ B4rY2+ B5yXyY +
$=
(9)
(10)
T , = ~ B , Y ,+ ~ B , Y , B1 = ( k 2
S
=Q=
01
tan2 0 2 ) - 2R(1 - k 2 ) tan
the measured voltage outputs from the anemometer are available. However it should be noticed t h a t when secondary flows exist ( R # 0 ) t h e usual simple s u m and differencing techniques so powerful for parallel flows n o longer can be applied. For the usual X array with cy1 = a2 = 45' the ratio of eq 17 and 18 gives 2
A similar equation can be developed for wire no. 2 We, = U C O S a 2 ( T 1+ T , + T,)", TI = ( k 2
_ from_(17) and (18) and the desired turbulent quantities, rx2, rr2 and Fy, from manipulations of eq 19 and 20 once
CY,
360 Ind. Eng. Chem., Fundarn., Vol. 14, No. 4, 1975
(11)
Once E1 and E2 have been measured and R determined from eq 21 then S1 and T I can be calculated and eq 17 or 18 used to determine U . The fluctuating quantities can be found by operating on eq 19 and 20 to obtain these relationships
3
FLOW
DIRECTION
-
I
I 1 B
1 A
I
i
MEASURING STATION f A O B
n
c
O
D
!4
I
1
1
C
D
1 \f
MEASURING STATIONS
Figure 2. Wavy surface and probe.
m
0
-jl
W
5
c
-4
B
01
02
03
04
05
06
07
08
09
07
08
09
I O
I I
Figure 4. Distribution of radial velocity. 40 w
9 IL
w
2
1
l
MEASURING STATIONS
301 MEASURING STATIONS 201
+
A
O
D
2
;
':: 1
01 0
I l l
' 01
02
03
04
05
06
10
r/R
Figure 5. Distribution of axial turbulence intensity.
-
- -
In the limit, R 0 these equations reduce to those given 0, k 0 , K1 = K2, by Sleicher and Hill (1969) and for R the equations become the cosine law. The solutions of eq 22-24 are clearly not possible with the analog equipment provided by commercially available anemometer equipment. In the experiments described below values of E ( t ) were recorded on FM tape, digitized, and the computations required to solve eq 21-24 were carried out readily using a special algorithm for that purpose. T h e effect of secondary flow on the computed values of the turbulence quantities can be deduced from the following: consider k = 0.5, a typical value for a film sensor and measurements in a field where R = 0.3. Equations 22-24 are stated below with the constants evaluated and compared with the case for R = 0, h = 0.5 as given by Sleicher and Hill (1969) (Sand H)
Eq 22:
(5)'
= 0.86'l14
([):
(:[(:)
S and H: = 1.0 -
Eq23:
(E)2 = 1.87 C[ 1.43
S and H:
-
= 2.78(:[
+
(2)12) -
( 2 ) ( :)I2) -
Figure 6. Distribution of radial turbulence intensity.
1.32($)]')
(a (:)I2) +
r/R
{~[l.lo(~)2
- 0.12(&
S and H: = 1.67{+[
- 0.92(92])
(z)2( )I': -
I t is evident that values of the turbulence terms calculated by ignoring the secondary flow would produce significant errors. Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
361
drops across wavy surfaces becomes clear. The existence of flow reversal and a recirculation eddy is indicated at station C. Percentage turbulence intensities in the axial and radial directions appear in Figures 5 and 6, and the turbulent shear stress in Figure 7. The shear is normalized by the friction velocity, U,, calculated from measured pressure drop along the pipe a t the wall using the average radius. The calculated distribution, using the minimum and maximum radii, is shown as dotted lines. The importance of the secondary flows as a mechanism for momentum transfer for flow over wavy surface is particularly clear from Figure 6. r/R
Figure 7. Distribution of turbulent shear stress.
Application Secondary flows and turbulence were measured for flow of air in a corrugated pipe of 8.986-in. mean diameter and 23 f t long having a near sinusoidal wall shape with wave amplitude of 0.219 in. and length 2.751 in. Air temperature was 77OF and measurements were taken 87 wavelengths downstream of the inlet with this location followed by approximately 3 f t of conduit. Pressure was essentially atmospheric at the discharge. A special X array was mounted as shown in Figure 2 where the location of measuring planes along the wave is also indicated. The probe and its support are shown approximately scaled to the size of the wavy surface. It was thus possible to measure within the trough of the waves. A series of runs at several flow rates were executed and details are provided by Chen (1973). Results for one flow rate are presented in Figures 3 to 7. The distribution of time average velocity in the axial direction appears in Figure 3. This velocity is independent of position along the wave except for locations within 15-20% of the surface. Time average radial velocity distributions are shown in Figure 4. Note that significant radial flows can be detected at distances 30% from the wall. When one considers the large momentum flux associated with very small radial flows, the origin of the very high pressure
Nomenclature e = fluctuating voltage output from linearized hot wire anemometer ( e = 0) E = time average voltage output from linearized hot wire anemometer k = constant in eq 1 K = constant relating output of anemometer to effective cooling velocity rx = ratio of fluctuating to time average velocity, u/U rs = ratio of fluctuating to time average velocity, ufU R = ratio of time average velocities in the y and x directions, V fU u = fluctuating component of velocity in the x direction U = time average velocity in the x direction u = fluctuating component of velocity in the y direction V = time average velocity in the y direction w = fluctuating component of the effective cooling velocity W e = instantaneous effective cooling velocity W I = instantaneous velocity vector LY = angle between the normal to the cylinder and the x direction fl = angle between the normal to the cylinder and the instantaneous velocity vector
Literature Cited Chen, W. N.. M.S. Thesis, University of Houston, 1973. Champagne, F. H., Sieicher, C. H.,J. fluidMech., 28, 153 (1967a). Champagne, F. H.. Sleicher, C. H.,J. fluid Mech., 28, 177 (1967b). Hill, J. C., Sleicher, C. H.,Phys. fluids, 12, 1126 (1969). McCroskey, W. J., Durbin, E. J., ASME Paper 71-WA/FE-17 (1971).
Receiued for reuiew April 2, 1975 Accepted July 7,1975
COMMUNICATIONS
The Effect of Purely Sinusoidal Potentials on the Performance of Equilibrium Parapumps
The equilibrium theory of Pigford is used to analyze the effect of purely sinusoidal velocity on separation in a closed parametric pump. The results are compared with separations obtained using square wave potential. It is shown that a square velocity wave can produce separation factors increasingly larger than with a sinusoidal velocity. The equations describing the system characteristics when both velocity and temperature are purely sinusoidal are also presented.
The underlying mechanism for separation in closed parametric pumps has been nicely explained with the publication of the equilibrium theory of Pigford et al. (1969). Elegant in its simplicity, the equilibrium theory has been the nucleus around which extensions to the continuous mode (e.g., Chen and Manganaro (1974)) have been built. Disper362 Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
sive effects were ignored in Pigford's model, hence ultimate separation is unbounded. Nonetheless, the theory provides a useful basis for comparison. One aspect of the equilibrium theory which appears to have been overlooked is the type of periodicity the velocity and temperature potentials take. In this work, we take the
