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MITED STATES \RTMENT OF 1MERCE

JUCATION

If"'

NBS TECHNICAL NOTE 563

Surface Pressure Fluctuations

Near an Axisymmetric

Stagnation Point

U.S.

VRTMENT

OF

MMERCE

National

Bureau

of

-. ndards

Lz

IU
NATIONAL BUREAU OF STANDARDS

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it A

UNITED STATES DEPARTMENT OF COMMERCE

Maurice H. Stans, Secretary

D^ NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Director

ft i ^O^i „„.

NBS

i,

TECHNICAL NOTE 563

ISSUED AUGUST 1971

Nat. Bur. Stand. (U.S.), Tech. Note 563, 67 pages (Aug. 1971) CODEN: NBTNA

Surface Pressure Fluctuations

Near an Axisymmetric Stagnation Point

Richard D. Marshall

Building Research Division

Institute for Applied Technology

National Bureau of Standards

Washington. D.C. 20234

V„u of "

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C, 20402.

(Order by SD Catalog No. C 13.46:563). Price .70 cents. Stock Number 0303-0896

CONTENTS

Page

List of Symbols v

1. Introduction 2

2. Theoretical Background 3

3. Experimental Techniques 18

4. Experimental Results 21

5. Conclusions 34

6. Bibliography 37

in

Symbol

A

C

C

D

CM

°p

cpf

D

D'

f

fAB(n)

Hz

K1} K

Lx , Ly, Lz , Lr

M

LIST OF SYMBOLS

Definition

Area

Contraction ratio

Drag coefficient

Virtual mass coefficient

Pressure coefficient

RMS pressure coefficient

Disk diameter

Maximum wake diameter

Coefficient of spatial, longitudinal

velocity correlation

Cross-spectrum function

f (n) = 4 f

Q rAB (T )cos(27rnT) dx. etc

Coefficient of spatial, lateral

velocity correlation

Symbol for cycles per second

Intensity, etc.

P U

Modified Bessel functions of the

second kind

Integral scales of turbulence in x, y,

z and r directions

Grid mesh size

Frequency

Total pressure, average

load per unit area

Mean pressure

Dimensions

2

L

L

L

,2 m_l _2 T Jf m L T ,F L T

L

L

i-l

FL"

FL

-2
Symbol

P

o

P

R

R(n)

R

AB

(n)

R(t)

Re

"AB

_u

:AB

t

U

U

u,v,w

J 2 *' 2 J~T

W

Wc

w

x,y,z

yAB

Definition

Stagnation pressure

Fluctuating pressure

Radius of disk

Narrow-band correlation function

Normalized form of cross-spectral

function

- ,,

f

AB fa> R«

(n) " V^~~

Space-time correlation function

Reynolds number

Radial distance

Spatial separation

Cross-correlation function

u

Dimensions

AB u

A

(t) * u

B

(t + T ) etc -

Time

Total velocity in x direction

Mean velocity in x direction

Velocity fluctuation in x,y and z directions

RMS of velocity fluctuations

Local mean velocity in z direction

Convection velocity

Free-stream velocity

Space coordinates

Lateral separation

FL

FL"

L

_2

L

L

l

2t" 2 ,f 2l" 4

LT

LT"

lt"

lt"

lt"

lt"

lt"

L

L

_1

_1

I _1

VI
Symbol Definition

(1) Non-dimensional function

(2) Normalization factor

Dimensions

*(n)

<f>(n)

(1) Non-dimensional function

(2) Angle

Angular separation of hot wire

and pressure tap

(1) Non-dimensional function

(2) Radial separation of hot wire

and pressure tap

Ndn-dimensional function

Mass density of air

Time delay

Spectral density function,

/°°$ (n) dn = 1

o Spectrum function

oo u 2 °° P 2

/ <£ (n)dn = u ,/ <J> (n)dn = p etc.

X*(n) Aerodynamic admittance fu

to Angular velocity

Subscripts

A Denotes point A

B Denotes point B

r Radial direction

3 Circumferential direction

Superscripts

P Denotes pressure

u,w Denotes velocity

FL hT

2

T

T 2m-1 T,2T -4m L T ,F L T

_1

vn
Symbol Definition Dimensions

( ) Denotes time- averaged quantity

u

2 = lim 1 r

T 2

4_ T— T I U (t)dt

J o

Other symbols are defined as they occur in the text.

VI 11
SURFACE PRESSURE FLUCTUATIONS NEAR

AN AXISYMMETRIC STAGNATION POINT

Richard D. Marshall

Structural Research Engineer

National Bureau of Standards

Washington, D. C.

Surface pressure fluctuations on a circular disk placed normal to a turbulent air stream have been investigated. Turbulence intensities of

approximately 10% were produced by a coarse grid installed at the testsection entrance. The turbulent field in the neighborhood of the disk

was homogeneous and nearly isotropic.

Experimental results indicate that existing linear theories which do

not consider distortion of the flow fail to predict the nature of surface pressure fluctuations on a bluff body. Only for wavelengths which

are large compared to the body do these theories yield satisfactory

results. A strong attenuation of the high frequency components occurs

as the flow stagnates. This is accompanied by a transfer of energy from

short to long wavelengths. The opposite effect is observed as the flow

attains a radial direction and approaches the edge of the disk. A neutral wavelength which undergoes little change in energy was observed.

Integral scales of surface pressure fluctuations are much larger than

the lateral integral scale of the free-stream turbulence.

Pressure-velocity correlations indicate the existence of two distinct

regions, an inner region in which correlations and optimum delay times

exhibit considerable change along the radius of the disk, and an outer

region wnerc t**o«« io i-ft-Ho der°ndence on radial distance. Maximum

values of the optimum correlations are found ±a the outer region. There

is qualitative agreement between the experimental results and theoretical predictions which consider the effect of vortex stretching.

Key words: Disk; pressure fluctuations; stagnation point; turbulence.
1. INTRODUCTION

Surface pressure fluctuations associated with and caused by turbulence have assumed considerable importance in recent years. This

interest has come about largely because of a growing awareness that

these pressure fluctuations play a significant role in the response of

buildings and similar engineering structures to atmospheric flows.

Gust loading of VTOL and STOL aircraft during takeoff and landing, vibration of boiler tubes, and the response of anemometers and wind vanes

also require a knowledge of surface pressure fluctuations and their

relation to turbulent velocities,,

Considerable effort has been directed toward the investigation of

wall-pressure fluctuations caused by turbulent boundary layers,, A comprehensive review of these investigations is presented by Bull, et al

[ 2] o Practically all of these studies have been carried out with the

plane of the boundary parallel to the mean flow direction. Although

they have contributed much to the understanding of aerodynamic noise

production and the structurp <*£ turmilent boundary layers, little information exists on fluctuating pressures acting on a boundary which is

not parallel to the direction of mean flow.

The problem of predicting these pressure pulsations is highly complex. A complete theoretical solution seems quite remote at this time

even if certain simplifying assumptions are made. A direct conversion

of velocity fluctuations into pressure fluctuations does not occur in

most real situations. The nature of the flow over the upstream portion

of a bluff body is governed to a large extent by the wake of the body
in the subsonic range. The wake, in turn, is influenced by body shape,

body roughness, Reynolds number, and free-stream turbulence. Even with

rather rash assumptions, theoretical solutions of the near wake for very

simple bodies are generally lacking as long as the fluid viscosity

remains finite. In the case of bluff bodies, the turbulent field is

severely distorted as it approaches and passes over the body, so that

assumptions of linearity generally break down. For these reasons one

must depend primarily on experimental observation for an understanding

of the mechanism by which turbulent flows produce surface pressure

fluctuations.

2. THEORETICAL BACKGROUND

Theoretical treatments of pressure fluctuations due to turbulence

have been restricted to homogeneous isotropic turbulence and certain

turbulent shear flows. In principle, the fluctuating pressure in a free

turbulent field can be derived from the Navier-Stokes equations and the

continuity equation as a Poisson equation

V (P - Pa) = "P ^ (UjUj)

(1)

3Xi 3Xj V }

where P is the ambient pressure and i and j correspond to the usual

tensor notation. The problem being considered here is complicated by

the fact that surface pressure fluctuations are due almost entirely to

turbulent velocity fluctuations in the stagnation region where vortices

undergo considerable distortion. To relate the surface pressure field

to the free-stream turbulent field therefore requires an understanding of

the stagnation process in a turbulent flow; i.e., the effects of fluid

3
strain on vorticity.

It is well known that turbulence intensities are drastically altered

when the flow is subjected to a strong acceleration. In the case of a

wind tunnel contraction, the longitudinal or streamwise component of

turbulence decreases while the lateral components tend to increase.

This phenomenon was first explained by Prandtl [13 ] . Prandtl assumed

the velocity fluctuations were due to vortices which were randomly

distributed. He reasoned that the velocity fluctuations in any given

direction were due primarily to vortices whose axes were normal to that

direction. Considering a contraction ratio C, the post-contraction

velocity is C times the precontraction value. A vortex filament oriented

in the flow direction is contracted by the factor — . If the strength

of the vortex remains constant (viscosity is neglected) , the angular

velocity must increase by an equal amount. The radius of the vortex in

the postcontraction region is reduced by a factor so the tangential

C

^

velocity is increased by the factor - = / C. Using a similar

/c~~

argument for vortices whose axes are normal to the flow direction, the

tangential or peripheral velocity decreases by a factor — . One would

then expect an increase in the transverse components and a decrease in

the longitudinal or streamwise component of turbulence as the flow passes

through a contraction. Although this simple linear relationship between

pre -and postcontraction values cannot be expected to hold in most real

situations (see Uberoi |21 ] ), it does provide a clear picture of the

effects of vortex stretching. The case of a contraction has been considered, but similar reasoning can be applied to expansions; i.e., decelerating

flow. The vortex filamaits would be subjected lateral stretching which

4
would intensify both the longitudinal and the transverse velocity fluctuations.

The two-dimensional case of vorticity amplification in stagnation

flow is the subject of theoretical paper by Sutera [ 19] . The model

assumes a Hiemenz flow with superimposed vorticity having the form of a

sinusoidal perturbation. Interaction of vorticity with the boundary

layer is also considered. An experimental verification of this theory

has been made by Sadeh Q.4] . Attenuation at high frequencies accompanied

by amplification at the low frequencies was observed. Both the theoretical and experimental results indicate the existence of a wavelength at

which the energy remains constant as the stagnation line is approached.

An exact solution of the incompressible Navier -Stokes equation at

an axisymmetric stagnation point with vorticity in the oncoming flow has

been presented by Kemp [7] . The free-stream vorticity considered by

Kemp varies linearly with the radial distance from the axis.

While it is convenient to think of the velocity fluctuations in

terms of discrete voritces, the problem being considered here requires

an approach based on the statistical concepts of turbulence. To obtain

realistic results, viscous effects and flow distortion would have to be

considered and under certain circumstances, the interaction between the

free-stream turbulence and the boundary layer.

A theoretical approach to the problem which fulfills some of these

requirements has recently been made by Deissler [4] . His analysis is

based on generalized two-point correlation equations which are obtained

from the incompressible Navier-Stokes and continuity equations. Shear

stresses are assumed to be absent and the flow is considered to be axisymmetric. The equations are solved by assuming the turbulence is weak

5
enough so that the terms involving triple correlations may be neglected.

The turbulence is assumed to be homogeneous in the transverse direction

and locally homogeneous in the longitudinal direction so that the scale of

turbulence can be considered to be much smaller than the scale of the

inhomogeneity

.

An additional assumption is that the turbulence is initially isotropic, but it is allowed to become anisotropic under the distorting

influence of the mean flow. The flow is also assumed to have uniform

normal velocity gradients which are allowed to vary axially but not transversely.

Deissler considers both accelerating and decelerating flows. His

analysis indicates that for a decelerating flow both the longitudinal

and lateral turbulence components decay as the stagnation point is

approached. If, however, the rate of normal strain is sufficiently high,

the turbulence intensities increase at a rate which more than offsets

the effects of viscous dissipation so that a net increase in turbulence

intensity is observed. The turbulence intensities increase without

bound as the longitudinal velocity component approaches zero. The original limitations placed on the analysis are violated at this point, so

that agreement with the real situation where the intensities remain

finite cannot be expected.

This effect of strain upon the turbulence components can be explained

more clearly by considering a random field of vortices as was done previously in the case of a wind-tunnel contraction. As these vortices are

transported along the stagnation streamline, the longitudinal components

of vorticity tend to decrease while the lateral components are increased

6
due to stretching. Vortex filaments whose axes were originally oblique

tend to align themselves in directions normal to the stagnation streamline due to the mean fluid strain in the transverse direction. Vortices

having this arrangement can then contribute to both the longitudinal and

transverse components of turbulence.

Deissler also considers the effects of strain upon the turbulence

spectra in a decelerating flow. The turbulent energy for longitudinal

and transverse components tends to concentrate at smaller wave numbers

than would be observed for simple decay without strain. This is also

the case for the variance of the transverse vorticity. The implication

here is that the larger vortices are amplified by the stretching process

as the flow approaches a stagnation point and vortices which are smaller

than some critical size tend to decay more rapidly under viscous action.

Although the response of a body to velocity fluctuations is not of

primary interest here, it seems appropriate to review some of the work

which has been done in this area, as the results have direct application

to the general problem of pressure pulsations.

The first rigorous approach to the problem appears to have been made

by Lin [11 ] . He considered the problem of a pendulum suspended in a

turbulent flow, the aim being to predict the motion of the pendulum in

terms of the free-stream turbulence by use of a statistical model. Lin

did not consider the distortion of the flow by the pendulum.

Liepmann [9 ] made use of the concepts of the stationary time series

in considering the response of aircraft wings to buffeting and was able

to relate the response spectra directly to the turbulence spectra. Flow

distortion was not considered since its effects were of a secondary nature

7
Liepmann's work was followed by the experimental work of Lamson [8] .

By instrumenting a segment of an airfoil placed in a turbulent flow, he

was able to obtain the spectrum of the fluctuating lift and thereby

establish an experimental verification of Liepmann's theory. Davenport

[3] extended Liepmann's theory to the problem of bluff bodies. Davenport

also assumed a stationary stochastic process, an assumption which has

been made throughout this study.

The simplifications which can be made in the case of streamlined

airfoils are not generally valid for bluff bodies. The one -dimensional

theory proposed by Davenport does not consider the problem of flow distortion directly but does put forth the fundamental concepts involved in

gust response. Assuming a homogeneous turbulence field, Davenport considers perturbations of the relationship between normal load per unit

area and velocity

P = \ U

2 C

D

(2)

where

P = P (t) = P + p(t)

and

U = U(t) = U + u(t)

Equation (2) can be separated into mean and time -varying components

P(t) = P + p(t) =\ p Tj2CD + p U u(t) C

D + 2 P u

2 (t) C

D

Neglecting second-order terms, the approximate relation between timevarying components becomes

p(t) *p U u(t) CD

. (3)

This relationship is valid of course only when u << U. If these components are indeed functions of time only, the one -dimensional normalized

8
power spectra are then related as follows:

neb (n) , nd> (n)

P U

To account for inertial forces, Eq. (4) is modified to read

/ C

2 (n) n * (n) . , D n <[, ( n) " * 2~~ J (5)

p cDU

The fact that the velocity fluctuations are caused by eddies of

finite size requires that some consideration be given to the spatial

correlation. Equation (4) is further modified to give

p , v C

2 (n) u, v n »^n> = 4 R

P

(n) -^ il^_Jnl

(6) _2

P ~D

C- U

x P/ N n<J> (n)

2 P

2 / \

D

(n)

2

/. n(J> u (n)

U

= R

p (n) a = X (n)

Since the force and velocity spectra can be determined experimentally,

Eq. (6) is rearranged in the form

(7)

2 P where X is termed the "aerodynamic admittance." Since R (n) is essentially a narrow-band correlation function, its magnitude must approach

unity for wavelengths which are large compared to the size of the body.

For very short wavelengths, R (n) must approach zero. The expected

behavior of C (n) is not quite so clear. For very long wavelengths,

where temporal derivatives are small, it should correspond to the steady

drag coefficient. For shorter wavelengths, it is reasonable to assume

that C (n) will tend to be larger than the steady drag coefficient, C .

However, short wavelengths imply a reduced lateral scale so that C (n) and

R

p (n) cannot be treated independently.

Observations of the dynamic response of low aspect-ratios plates

9
to turbulent flow have been made by Vickery [ 22] . Vickery used a coarse

grid to generate a turbulence field which was nearly homogeneous at the

point where the studies were conducted,, He also put forth a more sophisticated derivation for the aerodynamic admittance. Because of its

relevance to the present investigation, the main features of his work

are included here.

Assuming the turbulence to be homogeneous and isotropic, the diagonal terms of the two-point double correlation tensor are of the form

^"B

_2

u

2 (f-g) *? + g r

(8)

->- where f and g are scalar functions of the separation vector r connecting

point A at the origin with any other point B in the turbulent field and

having components x, y and z. For an incompressible fluid it can be

shown that

, x c , r 3f

gCr) = f +

2 ~r

It should be pointed out here that the coordinate system being

used in the present discussion does not correspond to the system used in

the main body of this paper. As defined here, u, v and w are the

turbulent velocity components in the x, y and z directions respectively,

x being the mean flow direction. The mean velocity is denoted by U.

10
By applying Taylor's hypothesis, the cross-correlation function

for the streamwise component is

uA(t) • u

B

(t + t) = u (f-g)

—~ -2 2 U T

~ + g

772 2^ 2 U T+yAB

(9)

where A and B are any two points in a plane normal to the flow with

separation y A „ D AB

Since f(r) is not specified, an exponential form is assumed,,

Hence

f (r) = e

g(r) =

x

1 -

2L

L

e x

(10)

and the expression for the cross-correlation function becomes

uA

(t) • u

B

(t + t)

where

a = Ut

u 2 = rAB

= U 1 -

2 /a'z + 3 Z

- /a z + B

z

(11)

and

'ABx

11
The cross- spectrum is now obtained by applying a Fourier transform

to the cross-correlation function

£Ib<<°> if. U -1C0T . rAB

6 dT < (12)

If the cross-correlation is symmetrical, then the imaginary part

of the cross-spectrum is zero and one obtains

iw^i (iwhere

cu L

n = u ~ and f

AB (n) T f

AB (w) x

If

6 = 3 / 1 + n

z

, the integral of Eq, (13) may be evaluated to give

*>>-*^H£ |K (?) - | K

o (6) (14)

where Ki and K are modified Bessel functions of the second kind.

o A normalized form of the cross-spectrum can be obtained by dividing

by the power spectrum in view of the assumption of homogeneity.

12
From Eq. (13), one obtains

oo

a u / \ 2 u

2 ( -a , v ^(n) = —— ) e cos (na) da =

2 u

s

Hence

= R^ (n) = 6K (6) -Ko ce)

JL

1 + n

z

(15)

Note that R» B

(n) is the equivalent of a narrow-band cross-correlation

function and is a function of 6 alone. Similar expressions are also

obtained for the v and w components but are not of interest here.

Considering the function , it is possible to simplify the normalized cross-spectrum function:

/ 1 + n

z =

For short wavelengths.

x ^

a) L

1 +

U

(16)

" yAB

2TM

yAB

U U

(17)

and for long wavelengths.

'AB L

* x

13
For short wavelengths, Eq. (15) then becomes

„u , , 2 TM ^E „ ( 2lrn yAB ) 2lr2"

2 y2AB „ |

2l,nyAB |

and for long wavelengths,

(19)

From the above analysis it is possible to make the following conclusions provided that the initial assumptions are correct and provided

Eq e (10) is a reasonable form for the correlation function,,

1 For wavelengths which are short compared to L , say X

2imL —— X u. — * 3 , R (n) is independent of the longitudinal scale

U

and is a function of frequency,,

2. For wavelengths which are long compared to L , say X

2irnL —— X u — < 1/3 , R (n) is independent of frequency and is a function

U

^

of the longitudinal integral scale L • X

3. RAT,(n) is a function of alone, and all values of

AB

R (n) should collapse on the same curve when plotted against AB

e .

The case of turbulent flow past a square plate whose dimensions are

small compared to the wavelengths of interest is now considered. For

the analysis to be valid, this assumption is essential since it allows

one to disregard the disturbances created by the plate.

14
The normal load per unit area on the plate is

P(t) = c

PU2 fr) + c P D

3U C t > ^ U; L

D 2

+ V D

at (20)

are where D is a characteristic dimension of the plate and C and C.

assumed to be constant. The auto-correlation function of the load per

unit area on the plate is

PA

(t) • pB

(t + t) = C£ p 2 u

2 uA

(.t) • u

fi Ct + t)

3u (t) 3u (t + t)

+ C£

P

2 ° n2 — -

3 t 3(t + t)

+ Q, Cp* U D uA (t)

3u

B

(t + t)

M ~DK A vw 3(t +t )

3u (t) +

Si V

2 u D 71 u

B

(t + T)

Now for homogeneous turbulence

uA

(t) • UgCt + t) = uA

(t - t) • u

fi (t) #

Hence,

3 Ul3 (t + x) u (t)

B 9u

A

(t "

T)

15
and

3uA

(t -x)

3(t - x)

72 9u

B

(t)

, rt

- -B

3 zuB (t + t)

3 Z (t + t)

Since the process is assumed to be stationary,

d

3(t + t) dx

and the auto-correlation function becomes

P t (t)-p.(t + x) "

C

D

'* V2 U

A

(t

>

• V+ T)

" C^ P

2 D^2 [uA

(t) .u

B

( t + x)]

(21)

Integrating Eq„ (21) over the area of the plate and taking the

Fourier transform of both sides of the equation gives

<|> F

(n) = C

2

p 2 U

2 An) // R^(n) ^A diL

B

+ 4tt 2 Cj p2 D

2 n2

c(, U

(n) // R^(n) dA

A

d^

<(» U

(n) C

2

p 2 U

2

X

2 (n) A2 (22)

where

X

2 (n) =

1

A?

4tt 2 C

2

1 +

M

<%

11 R

AB

(n) ^A "^

16
If G^ is small compared to C , the expression for the aerodynamic

admittance becomes

^-MKwdA.dA < 23> AB

Thus for wavelengths which are short compared to the longitudinal

scale of turbulence, X ( n) is primarily a function of the reduced

frequency — and plate geometry. For wavelengths which are long com- U

pared to L , the aerodynamic admittance is primarily a function of L

and D.

L

The range of values for ~i~ covered by Vickery [ 22] was approximately o 6 to 2.1. The observed aerodynamic admittance tends to be higher

L

than values calculated from Eq. (23) when ~J7 is small,, This is most

likely a result of neglecting C in Eq. (22) and ignoring the effects

of fluid strain on the turbulence. For values of -jT > 2, x becomes

n Jk practically independent of L and is a function of ——only. It seems X

L U

reasonable to conclude then, that for -j- <_ 1 , the simple linear theory

which relates the turbulence spectrum to the load spectrum tends to break

down and consideration must be given to the flow distortions caused by

the body.

17
3. EXPERIMENTAL TECHNIQUES

The experimental portion of this study was performed in the Meteorological Wind Tunnel, Fluid Dynamics and Diffusion Laboratory, Colorado

State University. The purpose was to determine the manner in which a

turbulent flow produces pressure pulsations over the face of a simple

body upon which it impinges. The wind tunnel is of the recirculating type

with a 9:1 entrance contraction and a 30 meter long test section. The

tesst-section entrance is approximately 1.8x1.8 meters, and an adjustable

ceiling makes it possible to regulate the streamwise pressure gradient

along the length of the tunnel.

In choosing a body on which to study pressure fluctuations, a

number of points were taken into consideration. The physical size of

the pressure transducers required a body whose dimensions would produce

a significant tunnel blockage as it was desired that the minimum spacing

between pressure taps be no more than about 10 percent of the body width.

For this reason, a disk was decided upon, and preliminary calculations

indicated that a diameter of 40 cm would produce a tolerable amount of

blockage. Another reason for choosing the disk was that the pressure

pulsations on the upstream face caused by instability of the wake would

probably be less severe as compared to a one-dimensional plate spanning

the test section. Subsequent investigations showed that contributions

to the pulsations over the front face of the disk from this source and

from tunnel noise were extremely small.

18
Available pressure sensing equipment required that the flow have a

high turbulence intensity,, In addition, it was desirable to have uniformity of both mean velocity and turbulence intensity across the tunnel

in the region where the investigation was to be carried out.

The most convenient method of generating high intensity turbulence

is by the use of a grid. Investigations carried out by Baines and

Peterson [ 1] on lattice-type screens indicated that intensities as high

as 20 percent could be obtained at about 10 bar widths downstream of the

lattice,, Vickery [22] used such a grid to simulate atmospheric turbulence

and obtained an intensity on the order of 10 percent at a distance of

eight mesh sizes downstream. His results indicated that the mean velocity

varied less than three percent across the tunnel at this point, and the

turbulence intensities were within 10 percent of the mean value. The

lateral scale was found to be o 6 times the bar width. Based upon the

results of these two studies and considering the scale of turbulence

required, a bi-planar grid was constructed of 38x6 mm aluminum bars

placed on 18 cm centers. The position of the disk relative to the grid

is indicated in Fig. 1.

Turbulence intensities and mean velocities at the center of a mesh

and centerline of a bar as a function of distance from the grid are presented in Figs. 2, 3 and 4. Based on these results, it was decided to

place the disk eight mesh sizes or 144 cm downstream of the grid since

the turbulence intensity at this point was considered to be the minimum

acceptable value.

19
Blockage due to the grid substantially diminished the dynamic

pressure that could be obtained with a given pitch and RPM setting of

the fan for clean-tunnel conditions. In order to obtain the largest

dynamic head and restrict the tunnel noise to an acceptable level, these

studies were conducted at a mean air speed of approximately 13 meters

per second. For the frequency range 0.3 to 300 Hz, this corresponded

to wavelengths in the range 0.1 to 100 times the disk diameter.

Mean velocity measurements were made with a pitot-static tube of

standard design. Velocity fluctuations and mean velocities near the

disk were measured by means of a constant-temperature hot-wire anemometer developed by Finn and Sandborn [5] . Measurements of mean velocities

by this method are subject to doubt when the turbulence intensity is

high as was the case very close to the front face of the disk. A

cross-wire probe was used to obtain lateral components of turbulence.

Two types of transducers were used to investigate the fluctuating

pressures on the disk; resistance type transducers incorporating unbonded strain gages and capacitance microphones. The transducers were

inserted into receptacles on the rear face of the disk and were connected to the front face by a 0.75 mm diameter orifice. The resistance

type transducers exhibited satisfactory response and lag characteristics

from to 60 Hz. The microphones had an attenuated response below 15

Hz and a flat response to 200 Hz. A detailed description of all equipment used in this study is presented in Ref. [12].

20
To determine the constraint placed on the wake of the disk by the

tunnel walls, a detailed survey of the wake at several stations downstream was carried out. Measurements were taken in both turbulent and

non-turbulent flows using the 40 cm disk and a disk having one-third

this diameter. If D is the disk diameter and D' is the maximum wake

diameter, the ratio D'/D for non-turbulent flow was found to be 1.74

and 1.78 for the 40 and 13 cm disks respectively. For turbulent flow

this ratio was approximately 1.54 for both disks. Mean pressure coefficients over the front face of each disk were in very good agreement

for both flows and it can therefore be concluded that wake blockage

produced little change in the flow characteristics over the front of

the disk.

4. EXPERIMENTAL RESULTS

In the following discussion, the origin of the coordinate system

is taken at the grid, with the z axis positive in the mean flow

direction. As is indicated by Figs. 2 and 3, the turbulence is nearly

homogeneous in the plane perpendicular to the flow direction at a distance

of twelve mesh sizes downstream of the grid. At a distance of sixteen

mesh sizes, it appears to become isotropic. The intensity of the longitudinal component of turbulence at eight mesh sizes downstream is

approximately 13 percent at the center of mesh and nine percent at the

center of a bar, based upon the local mean velocity. The corresponding

intensities of the lateral component are 10 and 11 percent. Figure 4

indicates that the velocity distribution becomes uniform at 10 mesh

sizes downstream.

21
The lateral correlation function for the longitudinal component of

turbulence is presented as a non-dimensional plot in Fig. 5. The data

presented in this figure were taken at the position normally occupied

by the disk which was eight mesh sizes or 144 cm downstream of the grid.

The lateral integral scale L was found to be 28 mm. Data obtained by

Vickery [22] using a grid similar to the one in this study and at the

same relative distance downstream, but with a Reynolds number about

twice as large, are also presented in Fig. 5. Although there is some

scatter in both sets of data, the agreement is rather good.

The longitudinal scale of turbulence was obtained from the autocorrelation function by use of Taylor's hypothesis [6] . The calculated

value of Lz is 67 mm. This function is plotted non-dimensionally in

Fig. 6. No data points are shown since the auto-correlation function

was obtained in the form of a continuous plot by a correlation computer.

Based on these results, the longitudinal integral scale is approximately

2.5 times the lateral scale at the normal disk location.

In Fig. 7 are presented the lateral narrow-band cross-correlations

or the real parts of the cross spectral densities of the grid turbulence

nLz for reduced frequencies — ranging from 0.06 to 0^90. Equation 15 is

W

also plotted in Fig. 7 for comparison. It is seen that the experimental w

values of R. ^(n) are consistently lower than the theoretical values due

AB

to the difference in the assumed and actual forms of f (r) . A damped

cosine curve has been fitted to the data in Fig. 7. This curve which has

t he form

- 7.2 (

2tt y r / \

> y cos 1.8tt/JL\ (24)

22
fits the data quite well except for very small values of separation yAp>.

It is seen that Ry,,(n) is related to the frequency in some fashion other

than is indicated by Eqs. (15) and (16). Also, the initial assumptions

are not valid at z/M = 8.

The spectral density function for the longitudinal component of

turbulence is shown in Fig. 8. If a descrete eddy of a certain size is

associated with a given frequency, the quantity W/n is then a measure of

the scale or wavelength. The power falls off rapidly for wavelengths

less than 30 cm per cycle with an attenuation rate of about nine db per

octave. The attenuation is constant down to wavelengths of 8 mm per

cycle where it becomes 16 db per octave. It is obvious from Fig. 8 that

5 the spectral function does not follow the - „ law at very high frequencies

and short wavelengths. This is no doubt due to the fact that the distance

from the grid is insufficient for the establishment of isotropic turbulence decay and to the fact that the Reynolds number is too low. Stewart

and Townsend [18 ] have shown that Re based on the mesh size should be at

6

lest of the order of 10 for the Kolmogoroff spectrum law to be valid.

The Reynolds number in this study was approximately 1.3 x 10 , based on

the mesh size.

The normalized spectrum of the longitudinal component at z/M = 8

(measured from the grid) is plotted in Fig. 9 along with the experimental

results of Vickery [22] as well as Taylor's one -dimensional spectrum [6] .

It has been assumed here that

f ,

, " yAB/L

y

f(r) = e

in calculating Taylor's spectrum. The anomaly associated with the

23
anisotropy of the flow is readily apparent in this figure.

The distribution of intensity of turbulence immediately upstream of

the disk with the grid installed is shown in Fig. 10. It should be emphasized that this is the turbulence intensity as measured by a hot wire which

is normal to a radial plane, i.e., normal to the local mean velocity vector.

The intensities are referenced to the free-stream velocity and z is

measured from the disk. There is a rapid decay along the stagnation

streamline (r=0) which begins at z/R =0.25 . in the region near the

disk (z < 25 mm) the turbulent energy increases with increasing r at a

fairly constant rate out to r/R - 0.5. Beyond r/R - 0.9 the turbulence

decays rapidly to a level at the edge of the disk which is approximately

equal to the free-stream intensity. This same trend is reflected by the

wall-pressure data of Fig. 12.

Properties of the flow along the stagnation streamline are indicated

in more detail in Fig. 11. The linear range is observed to extend out

to z/R - 0.8, and the ratio

TMTM

m was found to be 0.86 which is in good

Az/R

agreement with the non-turbulent value of 0.85. The intensity of turbulence based on the local velocity is also plotted in Fig. 11. The

fact that the hot-wire data are unreliable at low mean-velocities has

been noted previously. For this reason the data of Fig. 11 for z/R < 0.1

are open to doubt.

The nature of the flow in the region between the grid and the disk

does not permit a direct comparison with Deissler's theory [4] . There

is, however, qualitative agreement between theory and experiment. Based

on local velocity, the turbulence intensity passes through a minimum just

outside of the linear range and then increases at an increasing rate

24
through the linear range. Only a limited amount of the turbulence data

of Fig. 11 were subjected to a spectral analysis. Although the total

turbulent kinetic energy was found to decrease as the flow stagnated, an

increase in energy at long wavelengths was observed. At z/R = 0.1 and

r = 0, the turbulent energy was only 70 percent of the free-stream value

(disk removed) but the spectrum function at n/W = 0.8 meters " was

approximately 10 percent greater than the free-stream spectrum function

at the position normally occupied by the disk.

Pressure fluctuation intensities are plotted in Fig. 12 as a ratio

of RMS level to the free-stream dynamic pressure and are denoted by C ,.

Values of C

p f range from a minimum of 0.145 at the center of the disk to

a maximum of 0.182 near the edge. Following the same reasoning as was

used in obtaining Eq. (3), one obtains the relation

p

w w

Cpf

=

1/2 P W* = 2 C

p Wro (25)

At the stagnation point, C = 1 and C £ should then be twice the freestream turbulence intensity. The average intensity of the longitudinal

component of turbulence for free-stream conditions is 0.11 while the

average value of C near the stagnation point is 0.148. It is obvious

from Fig. 10 that the turbulence is suppressed as the flow approaches

the disk along the stagnation line. The intensity of turbulence which

satisfies Eq. (25) is I = 0.07, a value that is realized only very close

to the stagnation point. As will be indicated later, the velocity fluctuations which have the strongest correlation with the wall-pressure

fluctuations are those which occur at about 10 cm (z/R ~ 0.5) from the

stagnation point.

25
Radial correlations of the wall-pressure fluctuations are shown in

Fig. 13. The integral scales L for the reference points located at r

r R

are tabulated below:

L

r_ _r

R R

0.44

0.393 0.45

0.656 0.47

0.918 0.39

These correlations reflect the contributions of frequencies over the

range 0.4 to 200 Hz as do the circumferential correlations which are

presented in Fig. 14. A meaningful circumferential integral scale can

be obtained only from the data at r/R = 0.918 since the maximum separations for the other cases are insufficient to obtain zero correlation.

The integral scale for this case is 70 mm or about 80 percent of the

average radial integral scale. It is interesting to note that the average radial integral scale of the wall-pressure fluctuations is more

than three times the lateral scale of the grid turbulence at z/M = 8

(disk removed) . This implies that either the eddies are elongated as

the flow passes over the disk, or that the higher frequencies are attenuated, or that both occur simultaneously.

Narrow-band correlations were obtained using combinations of playback speed and filter band-width which resulted in an effective band

width of approximately one Hz. From an electrical analogy the real and

imaginary parts can be considered to represent the power in the in-phase

and 90 degree out-of-phase components respectively of two random signals.

The in-phase components were found to be approximately one order of magnitude larger than the out-of-phase components.

26
Comparing the integral scales as a function of frequency for the

data correlated with pressure fluctuations at the center of the disk, one

obtains _i , , x L (n) n , meters r

'

Wo,

8

1 6

3. 2

Lr (Wide band)

1.68

1.02

0.64

and for the circumferential correlation at r/R = 0.918,

- 1 L6 (n) n , meters ^ Wro L„ (Wide band)

0.8 2.09

1.6 1.48

3.2 0.65

It is seen that the integral scales do not vary directly with the

inverse of the frequency;, the scales associated with the low frequencies

being less than expected from a linear relationship between L(n) and

1/n.

Distribution coefficients defined in Ref. p.7] were calculated and

indicated that no strong periodic components were present in any of the

wall-pressure fluctuation data.

Spectral density functions of the wall-pressure fluctuations for

various values of r/R were also determined. The attenuation rate for a

given value of r/R was found to be constant for sufficiently large

values of —— and ranged from 19 db per octave at r/R = to 13 db per Woo octave at r/R = 0.918. The change in the spectral densities with r/R is

clearly demonstrated in Fig. 15. The constant 3.28 transforms the plotted

data of Figs. 15 and 16 into metric units. There is a continuous shift

in energy from the lower to the higher frequencies as r/R increases.

27
There is also a neutral frequency range which undergoes little change in

energy content. This frequency range is centered near 16 Hz and corresponds to a wavelength of about 75 cm per cycle based on the free-stream

velocity. The free-stream velocity has been used here rather than the

convection velocity due to the disparities in the latter. This will be

discussed in more detail later, but indications are that the convection

velocity over the radius of the disk is approximately equal to the freestream velocity. The energy increase across the disk is not reflected

by Fig. 15. This change amounts to about 50 percent but does not alter

the conclusions which have been drawn.

The aerodynamic admittance as defined in Eq. 7 is slightly modified

in relating the wall-pressure fluctuations to the velocity fluctuations.

In this study the aerodynamic admittance x is defined as follows:

XD

< r ' n> = * P( r > n) v $w (n) (26)

2 Figure 16 indicates the change in Xp

with r/R for various values of

n/(3.28 W ). It is apparent that the velocity fluctuations at the higher

frequencies contribute little to the wall-pressure fluctuations near the

stagnation point. To understand this, it is necessary to consider the

contributions that would be expected with a perfect transformation of

velocity fluctuations into pressure fluctuations.

If Cn

is replaced with C and C is assumed small in Eq. (21), the

expression reduces to

2 2 p(r,t) . p(r, t + T ) = C (r) p Wro w(t) • w(t + T ). (27)

p <=

Taking the Fourier transform of both sides, one obtains

2 2 ""2* _W

p° (r) $

F (r,n) = C (r) p W^ W $ (n) (28)

28
where

and

—i

P

,w

The aerodynamic admittance as defined by Eq. (26) then becomes

C*(r) (29) 4" - "2 ^r

where

-/ I = W

At the stagnation point the right-hand side of Eq. (29) becomes unity

if the turbulence and pressure fluctuation intensities are related as

indicated in Eq. (25). Due to viscous decay and the stretching of

vortex filaments, these simple linear relationships can be expected to

hold only for long wavelengths. The experimental value of the aerodynamic

admittance as defined in Eq. (26) was found to be 4.6 for n/(3.28 W^)

= o 01 meters „ Assuming that C f = 0.148 at the stagnation point, the

intensity of turbulence which satisfies Eq. (29) is I = 0.16; this same

intensity was observed in the region, of maximum velocity-pressure

correlation (z/R- 0.5). The aerodynamic admittance would be equal to

unity for the above conditions if the form indicated by Eq. (7) were used.

A value of 2.1 is obtained for the aerodynamic admittance if the freestream value of I = 0.11 is used

29
Insufficient measurements were taken to completely define the non-linear

process by which turbulent kinetic energy is transferred between different scales of motion. For this reason, spectral density functions

having unit area are particularly useful for illustrating the relative

distributions of energy in Figs. 15 and 16. Referring to Fig. 16 which

is based upon the turbulence spectrum at z/M=8 with the disk removed,

it is seen that there is a certain range of wavelengths which are relatively unchanged while the longer wavelengths are amplified and the

shorter wavelengths are attenuated. This neutral range corresponds

roughly to the neutral range of Fig. 15. As r/R increases, the contribution of the higher frequencies increases while that of the lower frequencies gradually decreases. At r/R = 0.6 there is a rapid increase in

the energy content of the high-frequency pressure pulsations, and this

trend continues out to the edge of the disk.

Typical space-time correlations of velocity fluctuations with wallpressure fluctuations are presented in Figs. 17-20. These correlations

were obtained so that the regions in the flow which contribute significantly to the wall-pressure fluctuations could be determined and are

defined as follows:

p(r,t) • Urr (n,Y,z,t - t)

R(n,Y»z,T) = fa (r,t) /u£ (n,Y,z,t^77 (30)

Correlations were obtained for a number of radial positions and

wire-transducer separations. These separations were vertical, denoted

by z; radial, denoted by n ; and circumferential, denoted by y • In this

expression, u represents the fluctuating velocity detected by the wire

which was positioned in a radial plane so that the wire axis was always

30
normal to the mean flow direction. For brevity, the correlation function

is denoted by R(t) in the figures.

The radial position of the hot wire was always equal to or greater

than that of the transducer in order to avoid problems of interference.

The correlations of Fig. 17 were most likely subjected to some interference since the pressure tap was immediately downstream of the hot

wire for this configuration. A positive time delay, T , corresponds to

a delay in the hot-wire signal relative to the pressure transducer

signal. Hence, if the optimum delay time is positive, the fluctuation

was first detected by the hot wire and at a later time by the pressure

transducer. Similar pressure-velocity correlations have been obtained

in turbulent boundary layers [16].

The optimum correlations and delay times are plotted in Figs. 21

and 22 along with additional data which are not included in the first

series of figures. Only those data for which the radial and circumferential separations are zero have been plotted in these two figures.

Figure 21 indicates that the maximum correlation between velocity and

pressure fluctuations occurs at z - 10 cm for all but values of r/R

approaching unity. The range of these maximum correlations is approximately 0.48 to 0.57. At large distances from the disk (say z/R > 1),

the correlations are only slightly less than the maximum values. It is

interesting to note that the region of maximum correlation corresponds

roughly with the region of maximum curvature of the streamlines associated

with potential flow. In the region close to the wall, the correlations

increase with increasing values of r/R for a given value of z . The

correlation tends to zero at the stagnation point and increases to a

31
maximum of 0.35 to 0.40 near the edge of the disk. It is difficult to

draw definite conclusions from Fig, 21 because of the scatter, but there

appear to be two definite regions involved, one region extending from

z = to z - 3 cm and another defined by z > 5 cm. In the first region,

the correlations increase, remain constant, or decrease with z . In

the second region the correlations increase to a maximum at z - 10 cm

and then slowly decrease with increasing z .

The two regions just defined for the optimum correlation coefficients

can also be applied to the optimum delay times which are plotted in Fig.

22. The minimum delay times are in general observed at z - 5 cm and

increase linearly with z beyond this point. For the region < z < 3 cm,

the delay times are zero at the stagnation point and decrease with increasing z for values of r/R up to about 0.6. Beyond this point the

delay times are practically independent of z .

Certainly the region adjacent to the wall is one in which the effects

of vortex stretching are predominant. There is also involved some interaction with the boundary layer. The boundary- layer thickness for nonturbulent flow was calculated from Homann's solution [15] and was found

to be approximately 0.1 mm. In this study, only the component of turbulence normal to the hot wire has been considered. It is quite likely

that the circumferential component of turbulence is just as important.

In fact the transverse component has been shown to be the most significant

one in the production of wall-pressure fluctuations by a turbulent

boundary layer [10, 20].

32
The outer region is bound to be less affected by vortex stretching

and consists of eddies which on the average are larger than those in the

inner region,, The observed velocity-pressure correlations are necessarily

associated with the larger eddies and therefore the lower frequencies.

This is so because of the spatial filtering of the smaller eddies as the

flow moves onto and over the disk. A small eddy which is detected by

the hot wire has either lost its identity or has decayed completely by

the time it is close enough to be detected by the pressure transducer,.

If only the large eddies are considered, then the quasi -linear theories

should be applicable. This is the reason for the very gradual falloff

in correlation as z increases. The larger the value of z , the

greater is the spatial filtering of the higher frequencies.

A majority of the pressure data in Figs. 17-20 was obtained with

microphones and was therefore subjected to a certain high-pass filtering.

To determine the effect of this filtering, a number of pressure-velocity

correlations were obtained using resistance-type transducers, i.e., full

band pass. The negative correlations had in general a larger absolute

value when obtained with the microphone, but the optimum correlations

were relatively unaffected by choice of transducer. The greatest deviation in optimum correlation was about eight percent; the correlations

obtained with the microphone were generally smaller.

The pressure-velocity correlations which have an angular separation

Y provide a measure of the circumferential scale. For small to intermediate values of r/R, the data indicate maximum correlations for z a 10 cm.

Near the edge of the disk, the maximum correlations occur at points much

closer to the disk, say z < 15 mm. This would suggest that vortices

33
undergo some stretching and tend to take on a circumferential orientation

near the edge of the disk.

Convection velocities were obtained from optimum delay times for

transducers having a known radial separation. Separation distances ranged

from 25 to 178 mm. These velocities are plotted as a ratio of the frees tream velocity in Fig. 23 and exhibit a considerable amount of scatter.

Investigations concerned with wall-pressure fluctuations beneath a turbulent boundary layer indicate a dependence of convection velocity on

frequency. This is explained by the fact that different frequencies

are associated with different regions of the boundary layer and therefore different local velocities. The data of Fig. 23 were subjected

only to the high-pass filtering mentioned above. In this case it is

difficult to define a convection velocity because the pressure fluctuations are caused not only by eddies being swept along the face of the

disk, but also by velocity fluctuations which occur far from the surface

where the flow pattern is entirely different.

5. CONCLUSIONS

The following conclusions can be drawn from this experimental

investigation of the pressure fluctuation correlations near an axisymmetric stagnation point:

1. The non-dimensional RMS pressure coefficient C _ is approximately

pf

twenty times that which is associated with wall-pressure fluctuations

beneath a turbulent boundary layer.

2. Fluid strain causes a strong suppression of turbulence along

the stagnation streamline, followed by strong amplification in the radial

34
direction. Viscous effects cause a drastic reduction of intensity at the

edge of the disk.

3. Simple linear theories fail to predict the nature of the pressure

fluctuations near the stagnation point. Long wavelengths are amplified

in the linear range while short wavelengths are attenuated. Experimental

results are in qualitative agreement with the uniform normal strain

theory of Deissler.

4. There is an increase in C _ with increasing r/R. The maximum

Pf

value of C

f occurs near the edge of the disk and is followed by a sudden

reduction at the edge.

5. Energy associated with pressure fluctuations is transferred from

long to short wavelengths as the radial distance increases. There exists

a wavelength which undergoes little change in energy.

6. Radial integral scales of the pressure fluctuations are practically independent of radial position and are about three times larger than

the lateral integral scale of the free-stream turbulence. The circumferential scale near the edge of the disk is approximately 80 percent

of the radial scale.

7. Pressure-velocity correlations indicate the existence of two

distinct regions, an inner region in which correlations and delay times

exhibit considerable change along the radius of the disk, and an outer

region where there is little dependence on r/R. Maximum values of the

optimum correlations are found in the outer region, and the correlations

decrease quite gradually with increasing z/R while the optimum delay times

increase linearly with z/R.

35
8. Although a wide range of wavelengths has been studied here, only

one value of turbulence scale and only one disk size were used. Extending

the range of L /R should be the subject of further study.

The writer wishes to express his thanks to Professor J. E. Cermak

for his able direction of this study and to Professor V. A. Sandborn for

specific suggestions concerning instrumentation and data analysis.

36
6. BIBLIOGRAPHY

1. Baines, W D. and Peterson, E. G., An investigation of flow through

screens. Transactions of the ASME, 73:467-477, July 1951.

2. Bull, M. K. , Wilby, J. F. and Blackman, D. R. , Wall pressure fluctuations in boundary layer flow and response of simple structures to

random pressure fields. AASU Report No. 243, 1963.

3. Davenport, A. G., The application of statistical concepts to the wind

loading of structures. Proceedings of the Institution of Civil

Engineers, 19:449-472, 1961.

4. Deissler, R. G., Weak locally homogeneous turbulence and heat transfer

with uniform normal strain. NASA TN D-3779, 1967.

5. Finn, D. L. and Sandborn, V. A., The design of a constant temperature

hot-wire anemometer. U. S. Army Research Grant DA-AMC-28-043-65-

G20 and NASA, NGR-06-002-038, CER 66-67CLF-36, Colorado State

University, Fort Collins, Colorado, March 1967.

6. Hinze, J. 0., Turbulence: An introduction to its mechanism and theory.

McGraw-Hill Book Company, Inc., New York, 1959.

7. Kemp, N. H., Vorticity interaction at an axi symmetric stagnation point

in a viscous imcompressible fluid. Journal of the Aero/

Space Sciences, 26:543-544, 1959.

8. Lamson, P., Measurements of lift fluctuations due to turbulence.

NACA TN 3880, 1957.

9. Liepmann, H. W. , On the application of statistical concepts to the

buffeting problem. Journal of the Aeronautical Sciences, 19:793-

800, 1952.

10. Lilley, G. M. and Hodgson, T. H. , On surface pressure fluctuations in

turbulent boundary layers. AGARD Report 276, 1960.

11. Lin,C. C, On the motion of a pendulum in a turbulent fluid.

Quarterly of Applied Mathematics, 1:43-48, 1943.

12. Marshall, Richard D. , Pressure fluctuation correlations near an axisymmetric stagnation point. Ph.D. dissertation, Colorado State

University, Fort Collins, Colorado, June 1968.

13. Prandtl, L», Attaining a steady air stream in wind tunnels. NACA

TM 726, 1933.

37
14. Sadeh, W. Z., An investigation of vorticity amplification in stagnation flow. Ph.D. dissertation, Brown University, Providence,

Rhode Island, June 1968.

15. Schlichting, Ho , Boundary layer theory,, McGraw-Hill Book Company,

Inc., New York, 1960, 4th Editionc

16. Serafini, J. So, Wall-pressure fluctuations and pressure-velocity

correlations in turbulent boundary layers. AGARD Report 453,

1963.

17. Simon, W. E. and Walter, L. A# , A new technique for determination of

cross-power spectral density with damped oscillators. Contract

NAS8-5322, Informal Report No. 18, Martin-Marietta Corporation,

Denver, Colorado, July 1967.

18. Stewart, R. W. and Townsend, A. A., Similarity and self-preservation

in isotropic turbulence. Phil. Trans. Roy. Soc. London, 243A:

359, 1951.

19. Sutera, S. P., Vorticity amplification in stagnation-point flow and

its effect on heat transfer. Journal of Fluid Mechanics,

21:513-534, 1965.

20. Tu, Bo Jo and Willmarth, W. W. , An experimental study of the structure

of turbulence near the wall through correlation measurements in

a thick turbulent boundary layer. University of Michigan,

Technical Report ORA-02920, March 1966.

21. Uberoi, Mo S., Effect of wind-tunnel contraction on free-stream

turbulence. Journal of the Aeronautical Sciences, 23:752-764,

1956.

22. Vickery, Bo Jo, On the flow behind a coarse grid and its use as a model of atmospheric turbulence in studies related to wind loads on buildingSo NPL Aero Report 1143, March 1965.

38
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61

ft U. S. GOVERNMENT PRINTING OFFICE: 1 971 ii8l -32i*/30

form NBS-114A (i-7i) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS-TN-5632. Gov t Accession No.3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Surface Pressure Fluctuations Near an Axisymmetric Stagnation

Point

5. Publication Date August 1971

6. Performing Organization Code

7. AUTHOR(S)

Richard D. Marshall

8. Performing Organization

10. Project/Task/ Work Unit No. 421.01

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

11. Contract/Grant No.

12. Sponsoring Organization Name and Address 13. Type of Report & Period Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) Surface pressure fluctuations on a circular disk placed normal to a turbulent air

stream have been investigated. Turbulence intensities of approximately 10% were produced by a coarse grid installed at the test-section entrance. The turbulent

field in the neighborhood of the disk was homogeneous and nearly isotropic.

Experimental results indicate that existing linear theories which do not consider

distortion of the flow fail to predict the nature of surface pressure fluctuations

on a bluff body. Only for wavelengths which are large compared to the body do

these theories yield satisfactory results. A strong attenuation of the high frequency components occurs as the flow stagnates. This is accompanied by a transfer

of energy from short to long wavelengths. The opposite effect is observed as the

flow attains a radial direction and approaches the edge of the disk. A neutral

wavelength which undergoes little change in energy was observed. Integral scales

of surface pressure fluctuations are much larger than the lateral integral scale

of the free- stream turbulence.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Disk; pressure fluctuations; stagnation point; turbulence.

18. AVAILABILITY STATEMENT

UNLIMITED.

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67

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