Effect of coriolis force on the turbulent boundary layer in rotating fluid machines

Signature redacted

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ii.

ACKNOWLEDGEMENTS

The author wished to express his gratitude to the members of the Gas Turbine Laboratory who made this work a rewarding experience.

'The author is particularly indebted to Professor Philip G. Hill whose assistance and guidance were invaluable, to Professor Edward S. Taylor, Director of the Laboratory, for his many helpful suggestions and to Mrs. Madelyn Euvrard for typing the manuscript.

The project was sponsored by the National Science Foundation Grant No. GI-220h.

111i

TABLE OF CONTENTS

ABSTRACT i

ACKNOWLEDGEMENTS ii

TABLE OF CONTENTS iii

NOMENCLATURE iv

LIST OF FIGURES v

I. INTRODUCTION 1

II. ANALYTICAL CONSIDERATIONS 3

III. APPARATUS

6

IV. INSTRUMENTATION 8

V. RESULTS AND DISCUSSION 10

(a) Mean Velocity 10

(b) Turbulence Intensities 12

(c) Skin Friction 13

VI. CONCLUDING REMARKS 14

VII. RECOMMENDATIONS FOR FUTURE STUDY 15

REFERENCES ₁₇

iv.

NOMENCLATURE

C f = Skin friction coefficient D = Width of channel

X, Y, Z = Orthogonal rotating coordinate system, see Figure lb U, V, W = Velocities in X, Y, Z directions respectively

u, v, w = Fluctuating velocities in X, Y, Z directions

P = Pressure

qT = Total turbulent energy

r, $,z = Cylinderical coordinate system

p = Density-of air

v = Kinematic viscosity of air

= Dynamic viscosity of air

= Potential defined as P + w2r2/2

6 = Momentum thickness

6* = Displacement thickness 6 = Boundary layer thickness

k = Unit vector in Z direction

w = Angular speed

T = Wall shearing stress Superscripts

()=

Vector quantity

( )

= Mean values Subscripts

(

= Free stream quantity

V.

LIST OF FIGURES

1. A General View of the Apparatus

lb. A Detailed Schematic of the Test Section

2. Data Recording Devices, Blip Ring and Bressure Transmitting Device

3. Measuring Probes, Traversing Mechanism and A General View of Hot-Wire Sets

4. Inlet Velocity Profiles

5. Inlet Turbulence

6. Static Pressure Distribution, w = 120 rpm

7. Velocity Profiles, w = 165 rpm

8. A Universal Plot of Velocity Profile at Station 4M, = 165 rpm

9. A Log-Log Plot of Velocity Profile at Station 4M, Suction Side, w = 165 rpm

10. Momentum Thickness vs. X . w = 165 rpm

11. Displacement Thickness and Boundary Layer Thickness, w = 165 rpm 12a. C vs. X, w = 165 rpm

12b. C f/C f vs. wX/U

13. U Fluctuations, w = 165 rpm 14. _{V Fluctuations, w = 165 rpm}

15. Apparent Shear, w = 165 rpm 16. Shear Correlation Coefficient

I. INTRODUCTION

Study of the effect of Coriolis forces on turbulent boundary layer was initiated in 1961 at the Gas;Turbine Laboratory under the sponsorship of the National Science Foundation. The purpose of the work is to obtain a better understanding of the general behavior of the turbulent boundary layer in rotating channels, e.g. the flow passages of turbomachines. The preliminary measurements had been reported in Gas Turbine-Laboratory Report No.

69

The measurements showed a substantial thinning of boundary layer on the pressure side while on the suction side wall the boundary layer thickness increased. Marked increase in turbulence intensity, Reynolds stress and

skin friction are recorded on the pressure side. These are the quantities that describe the general or gross behavior of the boundary layer and are

important parameters in predicting performance of radial flow pumps and compressors.

It is well known that transformation of an inertial coordinate system to a rotating coordinate system gives two-additional terms in the equations of motion. These two terms are centrifugal acceleration and Coriolis

accel-eration. In a duct flow, those two terms are balanced by pressure gradients.

Introduction of rotation will naturally introduce additional dimensionless parameters tothe'problem. These-dimensionless parameters are discussed in

Reference (4). To balance the Coriolis acceleration in a channel, the trans-verse pressure gradient is established. A similar situation exists in a curved channel flow where the'transverse pressure gradient is set up to balance the centrifugal acceleration. This pressure gradient affects the stability of the boundary layer. In the case of laminar boundary layer, Lord Rayleigh(5) established the stability criterion by means of a simple and elegant argument. Liepman has shown that Rayleigh's criterion is,

layer was stable while the concave-wall side was rather unstable. This

in-stability of the concave wall boundary layer resulted in early transition. Eskinazi and Yeh studied a fully developed turbulent boundary layer in a curved channel. They reported that on the convex side, there was a marked suppression of transverse fluctuations. Integral scales which are a measure

of large eddies of turbulence showed that the increase at the concave side is by a factor of four in comparison with that of the convex side for

v($, r)v(* + do, r) while for v(o, r)v(o, r + dr) the increase was about by a factor of two. Here $ is a direction parallel to stream direction while r is normal to the stream direction. These results appear to confirm the Rayleigh's criterion It is interesting to note here that Eskinazi and Yeh's

mean velocity measurements showed that a fully developed flow at the inlet of the curved channel eventually adjusted itself to form a large portion of irrotational flow in the central portion of the channel Hill and Moon(4 )

studied the turbulent boundary layer in a rotating channel and their prelim-inary conclusions on the production of turbulence and turbulence intensities were similar to that of Eskinazi and Yeh.

In the preliminary report, two dimensionless parameters were mentioned

Reynolds number UX

V

A parameter similar to wX

Rossby number U

where X is a distance from the inlet and w is angular velocity of the duct The Reynolds number is a measure of the-importance of the nonlinear-convec-tive acceleration force relanonlinear-convec-tive to the viscous force and the second para-meter is a measure of the importance of the convective acceleration relative to the Coriolis force. In the preliminary study, the range of -

investig-U

ated was limited for experimental reasons toO_ -<

8;

however, in view of the fact that in the case of radial flow machines the parameter

3,

can be as high as two, a new apparatus was built to produce ! to nearly two.

U

In the course of the present study, it became apparent that secondary flow in the channel had substantial-influence on the flow even at the plane of symmetry. A detailed study of the effect of secondary flow is now being conducted.

II. ANALYTICAL CONSIDERATIONS In the following, we let:

X, Y, Z be the rotating cartesian coordinates as shown in Figure lb.

UV,W be the mean velocity components along X, Y, Z respec-ively.

u,v , w be the turbulent velocity components along X, Y, Z directions respectively.

U, V , W be the total velocities.

The coordinate system is, as shown in Figure lb, attached to the channel and X is in the direction of flow, Y is perpendicular both to the direction of flow and the wall. Z is normal to both X and _Y. The equation of motion in a vector form on the rotating coordinate is:

U• V U =-2 w x U - V + v A U

where _{= +}

p 2

and r2 =x2 + y2 + z2

Here rotation vector is in the direction of Z only and writing

will reduce the rotation vector to a scalar and a unit vector in the direc-tion of Z.

In order to develop Reynolds equations, we will omit the Z component

h.

1) W < < U and all derivatives of W are small

2) Since the measurements are taken on a plane of geometrical-symmetry, it is expected that w and all cross-correlations containing w will be small.

It should be mentioned that some of the secondary flow study measurements without the geometrical symmetry showed that this position of zero stream-line curvature is extremely-sensitive to the flow conditions and that the point will deviate considerably. After following the procedure shown in Reference (4), the Reynolds equations become:

S + - +

V

+ a (uv)=

-2wV

-

-

2

x

++

ax

ax

ay

ay

p

ax

ay

2

(1)

3V

aV

a'i

= +- +ai + (uv) = 2wU - a y (2) The continuity equation is

+ = 0 (3)

ax 3y

The crosscorrelation uv is well known Reynolds' stress and is due purely to the turbulent motion in the flow. In the free stream region, equations

(1) and (2) can be approximated as

(A) 2x = + 1'

(4)

P,

ax

- 2U = - (5)

p ay

Equation (4) shows that the streamwise pressure gradient balances the cen-trifugal forces while the Coriolis force is balanced by the cross-streamline pressure gradient.

The total turbulent kinetic energy equation is

" 7 aIUV

+ 4w

1 + U Ta

2y 2ay ay--- ax ay

1 2 2) a2u

= (uax+ vy) (6)

5.

The first two terms are the transport of energy by mean motion, the third term represents the production of turbulence while the fourth and fifth are the convective transport. On the right hand side, the first term is the

rate of work done by the fluctuating pressure gradient and the last term is the combination of viscous dissipation and viscous transport. In equation

(6), the only term which differs from the two dimensional incompressible

flow is the production term. In the case of a channel flow without rotation,

the turbulence production is given by

-

au

uvay

By

It is apparent that because of the rotation, the production term is modified. It is interesting to compare the above two cases with a curved channel. In the curved channel, the production is also modified. Reference'(3) gives the production term as

-[ dU U uv

_Ldr

-- - -r

rJ

It is noted here that in the curved channel, the additional production of turbulence is due to the geometry. If the boundary layer thickness is-much smaller than the cross sectional width of the channel, then an approximation of the type

dU U

dr r

is valid. _{However, the additional production of turbulence in the rotating} channel is due to the rotation and is independent of the geometry.

For practical reasons, it is desirable to obtain readily-measurable quantities that adequately describe the gross behavior of the boundary layer without regard to the detailed variations in the boundary layer. Mager(T) integrated a _{general case of three-dimensional boundary layer equations for} the rotating fluid boundaries. _{The boundary layer equations are:}

6.

U + V -+ W = - + 2r + 2y + v 2

ax

ay

az

p

ax

ax

ay

_(Ta)

- 2r + 2wU= -(

ay

p ay

(Tb)

NW

ar + a

2

W

U - + W -_= - - + U)2r + 2

ax

ay

az

p az

az

ay

2

(Tc)

The continuity equation is

D+ 3V+ 1_= 0

ax

ay

az

(Td)

The above equations differ from Mager's in that they are specialized to the case where the axis of rotation is parallel to the wall. In the present investigation, we will assume that z direction momentum is small compared to x and y direction momentums. Integrating the above equations across the

boundary layer results in

aex

1

auW

;o

C

+ 20 + 6* + -

-ax

Ua

x

x

x

az

2

(8)

for x direction.

Methods of integration are shown in the Appendix together with defin-itions of terms. It can be seen from equation (8) that the first two terms are familiar Karman's integral terms. The third term is due to the cross

flow.

III. APPARATUS

A rectangular channel, as shown in Figure 1, with a dimension of 3" x

6"

x 72" constructed fro= plexiglass on the test walls and perforated metal

(40% open and covered with parachutecloth) on top and bottom walls is used as a test section. This channel is similar-to that reported in Reference

(4); however, the length is now increased to 72". The purpose of the

in-crease in length is to study the developing boundary layer, as generally

encountered in applications, and to increase the parameter w- to nearly two U

7.

The cross-sectional dimension has an aspect ratio of two. This is to minim-ize the secondary flow at the measuring stations. Furthermore, by bleeding some of the air in the boundary layer on the top-and bottom walls of the channel, it was hoped that there would-be further reduction in secondary flow. It was found that these precautions were not enough and that a consid-erable secondary flow did exist, which will be discussed in later sections. Hence, the top and bottom openings were completely sealed and all measure-ments were concentrated on the plane of symmetry as shown in Figure lb.

The test section is-mounted on a flat aluminum plate and the plate is rotated by an electrical motor connected by means of V-belts. The DC motor is rated at three Horsepower and the speed was controlled by means of a series of resistors connected-to the armature circuit. The center of rota-tion of the test secrota-tion and the flat plate is located at the mid-point.

For a source of air flow, a U. S. Navy Standard Vane-axial flow blower powered by a 3/4 Horsepower AC motor, is used. The capacity of the blower was rated as 1000 CFM at 3450 rpm. The blower was suspended from the

ceil-ing of the test room. The blower was stationary and located at the center of rotation of the test section. Sheet metal ducting was constructed to guide the flow into the inlet of the-test section. Honeycomb and screens

are placed in the guiding duct and inlet of the test section to give a flat velocity profile at the inlet for all speeds of rotation.

The test section and its mount was encased in a flat cylindrical box made of thin sheets of plexiglass with aluminum stiffeners. The purpose of

this encasing of the channel is to minimize disturbance around the channel while the channel is rotating and to give uniform bleeding of air from the top and bottom boundary layers in the channel. The channel was pressurized to cause a pressure drop across the walls.

bottom of the channel, as shown in Figure lb. The first station (lM) is located 1.5 ft. from the inlet and remaining stations are at a 1 ft.

inter-val 2M, 4M, 5M and 6M, in order. Station 3M is located at the center of rotation.

IV. INSTRUMENTATION

The pitot tube, shown in Figure 3, is constructed from hypodermic tub-ing flattened at the senstub-ing point to an outside thickness of 0.0005". The boundary layer thickness was between 1/4 inch and 1 inch. Meriam single tube inclined manometer was used to read the total pressure. A total of 12 openings with 0.03" diameter were made (six on each eall) on the test walls for the purpose of measuring the static pressure. These pressure measure-ments are transmitted from the rotating test section through a special pres-sure transmitting device shown in Figure 2. This device was developed at the Gas Turbine Laboratory. A detailed description of the device is given in Reference (4). _{For wall shear stress measurements, the Preston method} is used. _{The validity of the method is discussed in detail in Reference (4).} It was found that the Preston method gives wall shear stresses which check fairly accurately with turbulence measurements. In addition to Meriam man-ometer, a multi-tube manometer is used to read the static pressure at the walls.

The hot-wire anemometry consisted of a two channel transistorized

con-(4)

stant temperature equipment designed and developed by Kovasznay . The response is linearized to give the output voltage as a linear function of instantaneous velocity. _{Figure 3 shows the general view of-the hot-wire} anemometer. _{The probes, shown in Figure 3, are made of steel tubing and} sewing needles. _{Tungsten wire of 0.00015" diameter is used as a hot-wire.} There are two methods of attaching the wire on the needles; spot welding and

soft-soldering. The latter method-is used in the present investigation. Since the tungsten wire can not-be-soldered by itself, the wire is initially copper _{electroplated.} The process gives relatively easy way of producing

a well-defined wire length. The sensitive, or working part, of the wire is reasonably away from the supporting prongs to reduce the effect of distur-bance by the prongs and to minimize heat conduction to the needles. The gap between the needles was approximately 1/5 inch and the sensitive portion of the wire was approximately-1/10 inch long.,

Heathkit vacuum tube voltmeters are-used for the prupose of monitoring the voltage in the hot-wire circuits. A Hewlett-Packard vacuum tube

volt-meter Model 400C and Instrument Electronic voltmeter were used to obtain all rms readings. A Tektronic cathode-ray oscilloscope was used for visual

ob-servation of turbulence. Beed Electrical Instrument Co. DC'voltmeters are

used to obtain mean velocity from the hot-wire. The range of the voltmeters

was from 0 to 3 volts.

All the electrical signals were transmitted through a ten channel slip ring. The noise level of the slip ring is checked and is given in detail in Reference (4). The result of the noise test showed that there was prac-tically no noise generating from the contact _points. A general view of the

slip rings is shown in Figure 2. The mechanism of the traversing probes consisted of a selsyn motor and generator and reduction _gears. A general

view of the probe mounting mechanism is shown in Figure 3. The gear ratio is such that one turn of the generator advanced the probe a distance of 0.005 inch. This method of traversing of the probes made it possible to take a series of readings while the test suction is in _motion. All measurements

except the static pressure distribution discussed in the next sections w.ere made with the duct rotating at a speed of 165 rpm. The static pressure dis-tribution along the duct was taken with the-duct rotating at 120 rpm.

V. RESULTS AND DISCUSSION

(a) Mean Velocity

It was desirable that the velocity profile at the inlet of the test section be uniform for all angular apeeds of the channel. To accomplish this, the guiding duct system was pressurized by means of honeycomb and

screens. The guiding duct was

6"

x 6" and was constructed of sheet metal. The duct was contracted to 3" x

6"

near the inlet of the test section. To turn the flow in the guiding duct, aluminum turning vanes were used. Adjust-ment of the flow profiles at the inlet was accomplished by trial and error by altering the position of the screens. Figure 4 shows the result. Since

it was inconvenient to locate the measuring station at the inlet, the test of the velocity profile was made two inches downstream of the inlet. The results shown are for two angular speeds and it appears the velocity profiles are not affected by rotation. Figure 5 shows the measurements of the inlet

turbulence at the inl- station. Again, there appears to be no deviation between two measurements at zero angular speed and a speed of 165 rpm. At each measuring station, the static pressure was measured. Figure 6 shows the static pressure distribution along the duct. The wall static pressures were adjusted to give zero streamwise pressure gradient at zero rotation of the test section. However, as shown in Figure 6, there was a finite pres-sure gradient from the axis of rotation. This is due to the centrifugal

force by rotation of the channel. The pressure difference shown in Figure 6 is based on the atmospheric pressure. The pressure difference across the channel necessary to balance the Coriolis force is obvious. Figure 7 shows the measurements of velocity profiles with the channel rotating at a speed

WX

of 165 rpm. The range of - was from 0.48 to 1.92. The value 0.48 corres-ponds to the velocity profile at 1M while 1.92 corresponds to station 6M-It should be mentioned that station 6M was located only six inches upstrea from the outlet. _{A substantial increase in boundary layer thickness, as}

compared with stations Mand 5M on the suction side, seems to indicate that the velocity profile at station 6M was under the influence of the outlet. It can be seen from the plot that on the pressure side the boundary layer thickness remains nearly uniform for all stations while there is-a large increase on the suction side A typical velocity profile at station 4M is

replotted on a universal law plot0 Figure 8 shows the result. The

univer-sal- law of

5o6 log + 4_.9

is-chosen and are shown as dark lines using skin-friction coefficient as a parameter₀ The plot of 4M profile shows that on the pressure side the

pro-file coincides with the universal law fairly well while the suction side shows-rather large deviation Figure 9 is a replot-of the suction side

profile at station 4M. This result seems to indicate that on the- pressure

side the profiles fit the universal law fairly well while on the suction side the profiles are better fitted-by a'power law0 In the case of a

chan-el flow-without rotation, the-existenceof a-universal-law in-the vicinity of the wall for turbulenct flow is deduced taking into- account, apart from the small sub-layer xeat the'wall, only turbulent shearing stresses and is expected to satisfy only at large Reynolds numbers. For smaller-Reynolds

numbers where the viscous sub-layer exerts more influence on outer. layers, experiment leads to a power-lawof'the formshown'in'Figure 9. Here we see that for a given Reynolds number of 1.225 x 106, there are two types of

boundary layer existing. One of a universal law type at the6pressure side and the other a power law type at the suction side of the wall It appears

that this is an indication that on the-pressure side there is a substantial production of turbulence-while on the suction side there may be suppression

of'the turbulence. Figure-10 shows the-momentumthickness distributions

12. momentum thickness remains nearly constant. _{Figure 11 is the plot-of} calcul-ated _{boundary layer thickness and displacement-thickness. Although no} meas-urements in V and W velocities-are _{made in the present study, it will be of} interest to estimate the magnitude of flow in the third dimension. Since

the free stream velocity did not change much-in X-direction, it-will be au

assumed here that- is _zero. Figures 10 and 12a show momentum thickness and skin friction coefficients-respectively. _{It can be seen from the plots} that-on the pressure side _axx is nearly zero. It appears that shear stress on the pressure side of the wall is balanced by the cross flow in equation

36

(8). On the other hand, on the suction side - is rather large while the

ax

shear remains fairly _{constant. Hence, the cross flow is now balancing the} a0

momentum loss at the suction-side On the pressure side xz-is dominating

term and is of equal order of magnitude-to that of the skin friction coef-30

x

ison to the skin friction _coefficient. Therefore, it seems plausible to

aoe

assume that the cross flow term - is of the same order as C It apwe

az f

x

pears that the cross flow term contributes considerably and cannot be ig nored as being a second order term A detailed study it now being conducted

(b) Turbulence Intensities

Figures 13 and 14 show-the measurements of-u and v fluctuations at the plane of symmetry across the channel-for all _stations. The rotational speed

was 165 rpm. It is interesting to note here that the intensities near both walls do not change much for-all stations; however, the fluctuations in the

center portion-of-the duct-increase substantially at the downstream. The

fluctuations are spreading-from-the-suction side wall toward the-center portion which causes the increase-in fluctuations in the center-portion.

This is interesting, for Reference (3) showed that the spreading started from the pressure side It may be that the-secondary flow plays a role of

13.

convective transport fthe turbulence-from-the suction side to the-center portion of the flow. Theturbulenct shear measurements are shown in-Figure

15. There is a considerable increase-in the turbulent shear at the-pressure side compared-to that of the suctionside0 The suction side remains fairly

constant for-all stations. At station 1M and'2M, a;portion of the turbulent

shea-immediately outside of-the boundary layer is negative. This seems to indicate that the turbulence is being suppressed as-in the case of the cur-ved channel. Figure 16 shows the'shear'correlation coefficient for three

different cases. The solid line is theresult'reported in Reference (10). The dotted line belongs to the data-by Laufer and the point plotted are

for station 4M. On the pressure side, the data coincide withLaufer's reo sults fairly well while on the suction side the correlation is far below the other two cases shown. Here again, because of the unknown influence of flow in the z direction, further-study'is neededrto'verify the results.

(c) Skin Friction

Figure 12a shows the measured skin-friction coefficient. The measure-ment was done by means of Preston tube. Figure 12a isa plot-of the skin

friction coefficient against stations along the channel. As in the case of meanvelocity measurements, the channel was rotating with a constant speed

of 165 rpm. Since free stream velocity at all stations were nearly

con-WX WX

stant, the parameter was varied by changing X. The range of U was from 0.48 at atation lM to 1.92 at station 6M. For-the purpose of checking, the turbulent shearing stresses plotted-in'Figure,15 are extrapolated to the,, wall and assuming that the value at the wall is the wall'shear the skin

friction is computed. The value computed agrees fairly well with the Pres-ton tube measurements. Another check was made with the universal plot which uses-the skin friction coefficient as a parameter. It can be seen from Figures 8 and 12 that at station 4M the Preston tube measurements showthat

140 it is about 10% higher; however, considering possible secondary flow, the agreement is rather good.

As expected, the skin friction at-the pressure side is higher than on the suction side for all stations along the channel. On the suction'side, the skin friction coefficient decreases along the duct first and then it

increases. It should be mentioned here that in Reference (4) the skin fric-WX tion coefficient measurements are-limited in the range of-the parameter

-WX

from 0 to 0.8. When the range of u- is increased to nearly two, as shown

WX

in Figure 12a the skin friction coefficient begins to increase from U- 1.

On'the pressure side, the coefficients increasedmonotanically along the duct as _shown. Figure 12b is a plot of skin friction coefficient ratio.

C is the skin friction coefficient without rotation. All data points

0

shown are measured by means of Preston tube. The data were taken with the top and bottom porous and is presented to show the qualitative effect of the rotation.

VI. CONCLUDING REMARKS

1) Velocity profiles at the pressure side of the wall agree well with

the universal law and on the suction side the profiles show better agree-ment with a power law. The momentum thickness, displacement thickness and boundary layer thickness increased along the duct on the suction side of the wall while on the pressure side all remain fairly constant.

2) u and v fluctuations measured across the duct showed that near the walls the quantities remain fairly constant for all stations along the duct.

The spreading of u and v across the duct from the suction side of the wall toward the center portion. _{The turbulent shear measurements show that on} the pressure side there is considerable-increase along the duct and on the suction side the turbulent-shear remains rather uniform. The shear correl-ation is much lower than the straight-channel as well as the curved channel.

15.

3) Skin friction coefficient'increased along the duct on the pressure

side while on the suction side the-coefficient first decreased and then it

wX

increased from w- "' 1. Preston tube method of measuring the skin-friction agrees well with both the universal law and-extrapolated values of-turbulent shear at the wall.

4) An approximate estimate-shows that secondary flow affects the momen-tum balance _{considerably.} On the pressure side, the shear is practically

balanced by the cross flow-term in the momentum equation while on the suc-tion side the cross flow term appears to balance the momentum loss'. A de-tailed study is needed to confirm these findings.

VII. RECOMMENDATIONS FOR FUTURE STUDY

1) It is desirable to know-the relation between the parameter showing

oX

a degree of secondary flow with the-parameter -- To find this relation-U

ship, secondary flow measurements are essential. It is possible to measure the secondary component of the flow by means of hot-wires.

2) it would be interesting to study the type of secondary flow in rota-ting channels of different geometry. Generally, it is known that the shear flow in a curved duct results in secondary flow. Measurements show that the-secondary flow-is usually confined to the shear regions. In the case of rotating channel, it is possible that the flow in the axial direction helps to remove some of the flow from the pressure side and deposit it on the suction side.

3) With a known secondary flow in the rotating duct, it will be of in-terest to study the turbulence characteristics and compare them with that of non-rotating or rotating duct withouta secondary comporient..- This in formation iof importande when one, attemptstopredict thelperformance -of a given radial flow pump or compressor from data which is easily accessible.

16 4) Since the majority of radial flow pumps have diffusing chanels, it will be of practical importance to study a diffusing channel with or without curvatures,

17o

REFERENCES

1) Benton, G. S. - _{"The Effect of the Earth's Rotation on Laminar Flow in}

Pipes" - Jour. of Appl.Mechanics, Vol. 23, _{No. 1, pp. 123-127,}

March 1956.

2) Coles, D. - "The Law of the Wake in the Turbulent Boundary Layer"

-Jour. of Fluid-Mechanics, Vol. 1, pp. 191, 1956,

3) Eskinaze, S. and Yeh,H - _{"An Investigation on Fully Developed}

Tufbu--lent Flows in Curved Channels" - _{Jour, of Ae=. Sciences, Vol. 23,}

No. 1, pp. 23-34, January 1956.

4) Hill, P. G. and Moon, I-M, - "Effects of Coriolis on Turbulent Boundary Layer in Rotating Fluid Machines" - Gas Turbine Laboratory Report No. 69, _{Massachusetts Institute-of Technology, June 1962.}

5) Howard, I. N. - "Fundamentals of the Theory of Rotating Fluids" - Jour. of Appl. Mechanics, pp. 481-485, December

1963.

6) Liepmann, H. W. - _{"Investigations on Laminar Boundary Layer Stability}

and Transition on Curved Boundaries" - NACA Wartime-Report W-107,

August 1943.

7) Mager, A. - "Generalization of Boundary Layer Momentum Integral Equations to Three Dimensional Flow including those-of-Rotating

System"- 1 _{ACA TN2310, March1951.}

8) Schlichting, Ho -Boundary -Layer Theory - McGraw-Hill Book Co., New York, 1955.

9) Yeh, Hsuan _{- "Boundary Layer along Annular Walls in Swirling Floy"}

-Trans. ASME, Vol. 80, p. 797, 1958.

10) Yeh, H., Rose, W. G. and Lien, H. - _{"Further Investigations on Fully}

Developed Turbulent Flows in a Curved Channel" - The-Johns Hopkins University Mech. Eng. Dept.,!September _1956.

A-1

APPENDIX

The boundary layer equations are

U + V a+ W = + w2r + 2wV +vv

tx

y z p ax ax (Ta)

- o2r + 2WU = -

(Tb)

ay p 3y (7b)

with the continuity equation

__ + + = 0 (7d)

ax ay az

In the free stream -- is zero. The flow, therefore, is rotational. In

ay

order to evaluate the streamwise pressure gradient, equation (7b) is differ-entiated first with respect to X and integrated with respect to Y. The integration constant C which is a function only of X is evaluated at the

free stream. Differentiation and integration of equation (7b) gives

-w2r3 + 2w dy = - +

ax ax p ax ax (a)

Now, at the free stream, equation (Ta) gives au

U - = - 1 k+ w2r a + 2wV

00 ax

P ax

ax

00 Also, taking equation (a) to the free stream gives

w2 3r. -3C _{a- aC-r - --x} 1la 32 _{P -2 -- dy}w h3U

where h > 6.

Adding and solving for -- from the last two equations leads ax

3C

aiu

h

au

= - U -- +2w dy + 2wV

auav aw

Replacing by - -- from continuity, this equation is now

substi-ax ay z

tuted into equation (a)

= U 2 ar haw

- = U - 2WV + dy (b)

Pa

ax

Xx

a0

Iyaz

Equation (b) is now substituted into equation (Ta). This results in

U + V q+ W = + 2 aw aU

x y 3 Oax z y

A-2

Now, equation (c) is integrated across the boundary layer to the height of h. We will assume here that

h

2W h

J

dy

az y

is small compared to other terms in the equation since the term id of the order of 62 while others are of the order of unity. With definitions

1

h

6*= (U - U) dy x U 20 0 00+ xz f x J2h xz = _{( WUO - U) dy} 1h Sz " Z W2dy C D 0

Cfx

TOx _{yj au}

The final form of the integral equation in X direction is

^-+ °° alo 26 + 6* ]+ =6x - -f

FIGURE

I

AIR

guiding duct

6"

A9INLET

6"1

suction side

pre ssure

_2M

measuring stations

3

_5M

AXIS OF

ROTATION

X

Zcoordinate system

FIGURE

lb

r

4

W

7T7

0t

, VWi

Li

1.0

-8

6

INLET VELOCITY PROFILES

C

X

=

26rp

O

w=

0

A

w = 165 rpm

w

SUCTION

:-5

1-0

Y

PRESSURE

1-0

In_

inch

FIGURE

4

9Q

,fo

U

U

00

-2

0

-5

I

I

I

-4 __

5

u

U

x

100

Uw

-J

I

INLET TURBULENCE

a

=

=165rpm

C)

SUCTION

1-5

1-0

PRESSURE

1-0

Y inch

FIGURE

5

9

0

v

C

1-5

I I

I

I

I

STATIC

PRESSURE

0 w!120

A w=120

LM

RM

A

2

4

5

6

X

feet

rotation axis

FIGURE 6

2.0F

1.0

CL

.

0

1.0

9

I

I

I

I

6M - 5M 0-- _-4M 0(5 2M O0-- _-8O I M VELOCITY PROFILES

w=165

rpm 0-5 d - - _- -U SUCTION _PRESSURE 0 - Y Inch 0 I[50 _{[0 95} FIGURE 7

.01

.008

.006

.005

-00

-0

-.. ... .

11

lAj

1.0

.9

.8

.7

.6

.5

.4

.3

0 PRESSURE

Cf

as parameter

ASUCTION

_.0001

I I I B B I II I I I I I I I a

104

UNIVERSAL

PLOT OF VELOCITY PROFILE AT STATION

4M

w

= 165 rpm

FIGURE

8

u

/ 75.6

Log

+

4.9

U

U

.004

.003

.002

.21

.

I

0

102 a B . I B

a

a

M

I 1 1 16 1 1

I

I

a

a 11 1 a I

I

I

I

a

I a I I

.01

.008

.006

.005

A

I

I

I

I i

w = 165 rpm

I I I I I iii

a

a

ii

10i

LOG-LOG

PLOT OF VELOCITY

Uy

v'

104

PROFILE

AT

STATION

4M SUCTION

FIGURE

9

u

u

105

SIDE

L

.

.

.

.

.

.

.

.

,

,

,

,

,

_

I

I

_i

-

MOMENTUM

THICKNESS

w=165 rpm

-

= PRESSURE

SIDE

A= SUCTION

SIDE

.25[-A

0

2

0

3

X

feet

FIGURE

10

A

0

0

5

6

4

-c

C-) C

0

I

I

I

I

I

I

I I

i

I

I I I

suction side

pa e a I si

2.0

-5

X ft

2

BOUNDARY LAYER THICKNESS (8) AND

DISPLACEMENT

THICKNESS

(8)

FIGURE

II

8

8

8*

8*

0

A

a0

Oa

S=

165

for

for

for

for

I

rpm

suction side

pressure side

unit: INCH

I.0

-0'

C

I

3

4

6

I I I I

I

I

10

SKIN FRICTION COEFFICIENT

TOP AND BOTTOM SEALED

MEASURED WITH PRESTON TUBE

o

PRESSURE

SIDE

A

SUCTION SIDE

w=165

rpm

5-0

0

5

I23

4

5

6

X ft

FIGURE

12a

PRESSURE

0

0

0

V

Cf

Cfo-A

0

₀

EDAA

₀

v

A

0

.0

C

=skin friction coeff.

0

with w

= 0

-1.8

-1.6

-1.4

-1.2

1.0

0

Rx

=

5.73

x

10

5

V

0

0

9.9 x 10

5

1.09

x

106

1.29

x

1.54

x

porous top and

for bleeding

,

, ,

-0.8

0 a

-0.6

VV

1.0

A

10"

Iol

bottom

WX

U

09

V

0.4

-0.2

SUCTION

SKIN FRICTION COEFFICIENT

FIGURE

12

b

0

2

*,

2.10

RATIO

I '

I

U FLUCTUATION unit=FPS w =165 rpm O 6M 0 5 M 0 4M 0 2M 0 4 3 --2 u I M 0

SUCTION Y Inch PRESSURE

V FLUCTUATION unit=FPS S =165 rpm 6 M 11100 5M0 0~ 2M CI) IM I 5 SUCTION 1.0 _{Y inch} 1.0 FIGUR E 14 4 V 3 2 0 _1.5 PRESSURE

- iiv unit= PSF = 165 rpm 6M O O 5M - 4M ~ 2M~

-

--L

- O O

-

P-PRESSURE 0 Y inch FfGURE 15 0- 7-6. 4 3-2 0-1 -2- -3- -4- -5- -6-0 SUCTION 1.5 f.O0 I.0 15

SUCTION

PRESSURE

0

-/

0

-

0 00

0

/ / '

REF. 10

0

4M w=165rpm

0.8

1.0

CORRELATION

FIGURE

16

COEFFICIENT

1.0

0.8-

0.6-0.4

-0.2

-0.2

-0.4

-0.6

-0.8

-l.o

0

0.2

0.4

SHEAR

0.6

Y/D

i I

I

I

I I I I