600382007
      WAKE OF A BLOCK VEHICLE  IN A
        SHEAR-FREE BOUNDARY  FLOW
 An Experimental  and  Theoretical Study
                   by
            Robert E.  Eskridge
                   and
            Roger S. Thompson
    Meteorology and Assessment Division
Environmental  Sciences Research Laboratory
    U.S.  Environmental Protection Agency
      Research Triangle Park, NC  27711
 ENVIRONMENTAL  SCIENCES  RESEARCH LABORATORY
     OFFICE OF  RESEARCH  AND DEVELOPMENT
    U.S.  ENVIRONMENTAL PROTECTION AGENCY
      RESEARCH  TRIANGLE  PARK, NC  27711

-------
                                 DISCLAIMER

     This report has been reviewed by the Environmental  Sciences  Research
Laboratory, U.S. Environmental  Protection Agency,  and approved for pub-
lication.  Mention of trade names or commercial  products does not constitute
endorsement or recomnendation for use.
                                 AFFILIATION

     Robert E. Eskridge is a research meteorologist in the Meteorology and
Assessment Division, Environmental Sciences Research Laboratory, U.S. Environ-
mental Protection Agency, Research Triangle Park, North Carolina.  He is on
assignment from the National Oceanic and Atmospheric Administration, U.S.
Department of Commerce.
                                      11

-------
                                   ABSTRACT

      The wake of a moving vehicle was simulated using a specially-constructed
wind tunnel with a moving floor.  A "block-shaped" model vehicle was fixed in
position over the test-section floor while the floor moved at the freestream
air speed to produce a uniform, shear-free, approach flow.  This simulates an
automobile traveling along a straight highway under calm atmospheric conditions,
     Vertical and lateral profiles of mean and fluctuating velocities and
Reynolds stresses in the wake of the vehicle were obtained using a hot-film
anemometer with an X-probe.  Profiles were taken at distances of 10 to 80
model heights downwind.
     A momentum type wake was observed behind the block-shaped vehicle.  The
wake does not have a simple self-preserving form.  However, it is possible to
collapse the velocity deficit with one length and one velocity scale.
     Two new theories for the velocity deficit are compared to the theory of
Eskridge and Hunt (1979).  A theory which considered a height-dependent eddy
viscosity was found to fit the data best.
     Length and velocity scales were found for the longitudinal variation of
the turbulent kinetic energy.  The lateral variation is described by a two-
dimensional numerical fit of the crosswind variation of the data.
                                       111

-------
                                  CONTENTS
ABSTRACT	ill

FIGURES	vi

TABLES 	  x

SYMBOLS	xi

ACKNOWLEDGEMENTS 	  xv

     1.  INTRODUCTION	    1

     2.  SUMMARY AND CONCLUSIONS 	    5

     3.  APPARATUS	    7

         3.1  The wind tunnel	    7
         3.2  Velocity measurement 	    8
         3.3  Model  vehicles 	    8
         3.4  Empty wind tunnel  flow characteristics 	    9
         3.5  Wind tunnel  blockage effects and
              Reynolds number independence 	    9

     4.  WAKE THEORY AND MEASUREMENTS OF VEHICLE WAKES 	    13

         4.1  Review of vehicle wake theory  	    13
         4.2  Verification of the theory 	    15
         4.3  Modified wake theory 	    16
         4.4  Height dependent viscosity wake theory 	    17
         4.5  Turbulent kinetic energy terms in the wake	    18
         4.6  Reynolds stresses in the wake	    22

REFERENCES	    24

APPENDIX A  Control  volume analysis	    26
APPENDIX B  Multi-scale wake theory	    37
            B.I  Derivation of the multi-scale wake theory 	    37
            B.2  Longitudinal variation of the wake velocity
                 deficit solution	    41
APPENDIX C  Height-dependent viscosity wake theory 	    43
APPENDIX D  Simplified height-dependent viscosity wake theory  	    46

-------
                                    FIGURES


Number                                                                  Page


  3.1       The wind tunnel  	   49

  3.2       Scale model  vehicles 	   50

  3.3       Empty tunnel, mean velocity profiles for four longitudinal
            positions on center!ine 	   51

  3.4       Empty tunnel, mean velocity profiles for three lateral
            positions at a distance of 1  m from the test section
            entrance	   52

  3.5       Empty tunnel, turbulence intensity profiles for three
            positions at a distance of 1  m from the test section
            entrance	,	   53

  3.6       Locations of pressure measurements for determining
            blockage effects 	   54

  3.7       Spanwise variation in C   for various blockage ratios,
                                   pb
            AM/AT	   55

  3.8       C   as function  of Reynolds number for various A^/A^	   56
             "b
  3.9       Variation of C_   with blockage ratio 	   57
                          pb

  3.10      Vertical profiles of deficit velocity at x/H = 15, y =  0
            for 1/8 and 1/32 scale vehicles 	   58

  3.11      Lateral profiles of deficit velocity at x/H = 15, z/H = 0.5
            for 1/8 and 1/32 scale vehicles 	   59

  3.12      Vertical profiles of fluctuating velocity component at
            x/H = 15, y = 0 for 1/8 and 1/32 scale vehicles 	   60

  4.1       Vertical profile of velocity deficit at x/H = 15, y = 0
            for 1/8 scale block vehicle and EH theory 	   61

  4.2       Vertical profiles of velocity deficit for 1/32 scale
            vehicle y = 0 	   62
                                       vi

-------
Number                                                                Page

  4.3  Decay of maximum velocity deficit with  downwind
       distance	        63
  4.4  Rate of growth of the vertical  length scale  as determined
       from velocity profiles 	        64
  4.5  Smoke release showing slow growth of the  wake  	        65
  4.6  Measured Reynolds stress divided  by velocity shear  as
       a function of downwind distance 	        66
  4.7  Theoretical  curves and data for vertical  profile  of
       velocity deficit for x/H = 30 and y = 0 	        67
  4.8  Theoretical  curves and data for lateral profile of
       velocity deficit for x/H = 30 and z/H = 0.5  	        68
  4.9  Lateral  profile of velocity deficit for three  longi-
       tudinal  positions at constant c 	        69
  4.10 Theoretical  curve and lateral  profiles  of velocity  deficit
       at constant 5 using similarity coordinates 	        70
  4.11 Theoretical  curve and vertical  profiles along  centerline
       of velocity deficit using similarity coordinates  	        71
  4.12 Lateral  variation of the correlation of the  fluctuating
       velocity component, v' 	        72
  4.13 Length scale determined from fluctuating  velocity
       component v'  	        73
  4.14 Correlation of the fluctuating velocity component in
       similarity coordinates 	        74
  4.15 Longitudinal  variation of the maximum values of the
       turbulent kinetic energy components u'2 and  v'2 together
       with least squares linear fit 	        75
  4.16 Cross section of normalized turbulent kinetic  energy
       component u'z  with  analyzed  isolines  of  the distribution
       at x/H =  30  	        76
  4.17 Cross  section  of normalized  turbulent kinetic energy
       component v^with analyzed isolines  of the  distribution
       at x/H = 30 	                                77
                                     Vii

-------
Number                                                                Page

  4.18      Cross section of normalized turbulent kinetic energy

            component w'2 with analyzed isollnes of the distribution
            at x/H = 30 	    78

  4.19      Cross section of F  (x,w)  with analyzed isolines of
                              \j>

            the distribution of FC 	    79

  4.20a     Measured center!ine vertical variation of the turbulent
            kinetic energy component u'2 at various downwind
            di stances 	   80
  4.20b     Measured center!ine vertical variation of u'2 plotted in
            TKEC similarity variable compared to eq.  4.21 (solid
            line) 	    81

  4.21 a     Measured center!ine vertical variation of the turbulent

            kinetic energy component v'2 at various downwind
            di stances 	   82

  4.21b     Measured centerline vertical variation of v'2 plotted in
            TKEC similarity variables compared to eq. 4.21 (solid
            line) 	   83

  4.22a     Measured centerline vertical variation of the turbulent
            kinetic energy component w'2 at various downwind
            distances 	   84
  4.22b     Measured centerline vertical variation of w'2 plotted in
            TKEC similarity variables compared to eq. 4.21 (solid
            1 ine) 	,	   85

  4.23a     Measured lateral variation of the turbulent kinetic energy
            u'2  at various downwind distances 	   86
  4.23b     Measured lateral variation of u12 plotted in TKEC simi-
            larity variables compared to eq. 4.21 (solid line) 	   87

  4.24a     Measured lateral variation of v'2 at various downwind
            distances 	   88
                                      vm

-------
Number                                                                Page


  4.24b     Measured lateral  variation of w'2 plotted in TKEC simi-
            larity variables  compared to eq. 4.21 (solid line)	  89
  4.25a     Measured center!ine vertical  variation of uV at
            various downwind  distances 	  90
  4.25b     Measured center!ine vertical  variation of uV in EH
            similarity coordinates 	  91

  4.25c     Measured center!ine vertical  variation of u'w' in TKEC
            similarity coordinates 	  92
  4.26a     Measured lateral  variation of u'w1  at various downwind
            distances ................................................   53

  4.26b     Measured lateral  variation of u'w'  in EH similarity
            coordi nates  ..............................................   94
  A-l        Control  volume for determining the couple ................   95

  B-l        Example  of cubic spline fit to velocity profile at
            x/H = 10,  y = 0 ..........................................   96

  C-!        Example  of velocity spectra at x/H =15.   Spike near
            20 Hz is due to a vibration from the moving belt .........   97

  C-2        Vertical variation of the eddy viscosity determined
            from the spectral data with error estimates ..............   98

  C-3        Vertical variation of the eddy viscosity determined
            from the Reynolds stress u'w1  	  99

-------
                                    TABLES
Number                                                                 Page
  1.1          A Summary of Theoretical and Experimental Deter-
               minations of Velocity and Turbulence Characteristics
               of Wakes behind Two and Three-dimensional Objects. ...  3
  3.1          Drag Coefficient as a Function of Blockage	9
  4.1          Polynomial Two-dimensional Fit to Turbulent Kinetic
               Energy Function F	20

-------
                                    SYMBOLS
A                 — constant, determined from the couple, giving the strength
                     of the wake
Ajyj                -- cross-sectional area of model
Ay                — cross-sectional area of wind tunnel
a-|, 82, 83        — constants in turbulent kinetic energy equation

C                 -- couple on the vehicle
Cg                -- drag coefficient
C                 -- pressure coefficient on back face of model
 pb
D                 — drag on the vehicle
f                 -- function giving vertical  and lateral variation of velocity
                     deficit
F                 — function describing lateral and vertical variation of
                     turbulent kinetic energy terms in original (EH) theory
FC                — function describing cross-sectional variation of turbulent
                     kinetic energy terms
F                 -- force vector
H                 — height of the vehicle
H(?)              -- normalized function describing vertical variation of
                     eddy viscosity
H*(x,z)           -- function describing vertical variation of eddy viscosity
                                        XI

-------
i                 <-- unit vector 1n x or x, direction
->-
j                 — unit vector in y or x2 direction
->•
k                 — unit vector in z or Xg direction

l(x)              — vertical scale length of velocity deficit
V(x)             — vertical scale length of turbulence wake
i                 — Eulerian integral length scale

M(a,b,z)          -- functional form of a type of power series
->•
n                 — outward directed unit normal
p                 -- perturbation pressure
P.                 -- static pressure measured on rear of vehicle
P                 -- total pressure
P                 — free-stream static pressure
  r
p                 — static pressure measured with pi tot tube
->•
q                 -- three-dimensional velocity vector
Re                -- Reynolds number
s^               — fluctuating rate of strain
S.j.               — mean rate of strain

u                 -- velocity deficit behind the vehicle defined so that u ^ 0
umav              — maximum velocity deficit
 max
U                 -- local mean speed
                                       xii

-------
U                 — free-stream speed
 oo


u'2               -- x-component of turbulent kinetic energy

uV, u'w1, etc.  -- Reynolds stresses

v                 — y-component of perturbation velocity
v'2               -- y-component of turbulent kinetic energy



W                 — width of the vehicle

w                 -- z-component of perturbation velocity
w'2               — z-component of turbulent kinetic energy
(x, y, z)         -- right handed coordinates, with x-axis or x,-axis
                     directed along axis of the wake

\X-ij X* , X«/


x", y", z"           -- similarity coordinates used in velocity deficit
                     solution

X-| , X2> Y, , Y£    — arbitrary points used in control volume analysis
( )x              — indicates partial derivative with respect to x

a, e              -- constants relating the decay in the wake to the growth
                     of the length scale

Y                 -- constant in assumed form of eddy viscosity

6                 -- height of boundary layer in wind tunnel

S                 — similarity coordinate, z/l(x)

n                 — similarity coordinate, y/l(x) or y/[xW Hl(x)]

x                 -- constant relating transverse eddy viscosity to the
                     vertical eddy viscosity


                                       xiii

-------
A,             — constant in original  velocity deficit theory


a              — standard deviation of velocity deviations

v              -- kinematic viscosity

vy             — eddy viscosity

p              ~ density of dry air


Txy            — shearing stress acting in x direction along a surface
                  normal  to the y direction


X              — similarity coordinate, x = y/[W (x/H)0'4]

fj .            -- constants in polynomial surface fit


oj              — similarity coordinate, u> = z/[H (x/H)0'4]
                                      xiv

-------
                               ACKNOWLEDGEMENTS

     The work reported 1n this paper was greatly Influenced by the following
people; we thank:
     Rex Brltter for his help in designing the wind tunnel  and discussions
concerning the results,
     Ian P. Castro for his assistance in the tunnel blockage effect and
Reynolds-number-independence work,
     Julian C.R. Hunt for his suggestions on alternate theoretical approaches
to the original theory and for his review of this paper,
     Myron Manning for his assistance in constructing and modifying the moving
floor wind tunnel  used in this study,
     Mike Shipman  for generating new and modifying existing computer programs
for special purposes associated with the collection and reduction of data, and
     The remainder of the staff at the Fluid Modeling Facility for their many
comments and suggestions.
     Their generous assistance is deeply appreciated by the authors of this
report.
                                      xv

-------
                              1.  INTRODUCTION

     The dispersion of automobile exhaust near highways is of much current
interest.  Mathematical models are needed that will accurately predict con-
centration fields. This study on the structure of the wakes of vehicles was
undertaken at the U.S. Environmental Protection Agency's Fluid Modeling
Facility to provide information to be used in numerical models of automobile
exhaust dispersion.
     A wind tunnel with a test-section floor that moves at the freestream air
speed was constructed to generate a uniform (constant speed with height)
approach flow.  Scale model vehicles with rotating wheels were held in posi-
tion in the test section by guys attached to the test-section walls.  A vehicle
fixed in a uniform flow over a moving floor is aerodynamically equivalent to
a vehicle moving down a straight highway through a calm atmosphere.
     Automobiles and scale model vehicles have wakes which can be characterized
as momentum wakes which contain an organized vortex pair aligned with the axis
of the wake.  In this paper we are interested in only the momentum wake and
therefore generally discuss only those data and theories applicable to momentum
wakes.  To create a "pure" (no mean swirl) momentum wake, block-shaped vehicles
of appropriate heights and widths have been used.  It should be noted that
momentum wakes contain vortices, but in time-averaging the flow the vortices
disappear.
     A momentum wake's character is determined mainly by the flow field in
which it is embedded.  Important cases are:  flows which are free of limiting
walls, and flows which are influenced by a boundary.  Free flows may be sub-
divided further into shear flows and shear-free flows.  In addition, wakes in
the flows can be subdivided into laminar and turbulent wakes.  The classical
results for unbounded laminar wakes are given by Goldstein (1938) and a rec-
ent theory has been developed by Hunt (1971) for surface-mounted objects.
Hunt showed that the strength of the wake for surface-mounted objects is deter-
mined by the couple rather than the drag.

-------
     Hunt's work was extended by Counihan £t al_.  (1974)  to describe wakes
of two-dimensional objects in a turbulent boundary layer.   Castro (1979)  has
shown that this theory is inapplicable when the wake height approaches that of
the boundary layer.  However, the wakes of very small  two-dimensional  objects
didn't agree with the theory as well as those behind large two-dimensional
blocks.  Castro stated that it is unlikely that a constant eddy viscosity
model will describe the complex flow in this type of wake.
     Theories for wakes behind three-dimensional  objects in turbulent  boundary
layers have been developed by Hunt (1971, 1974),  Lemberg (1973) and Hansen  and
Cermak (1975).  Eskridge and Hunt (1979), hereafter referred to as EH, have
presented a model of a three-dimensional wake near a moving surface (no shear
in the approach flow).  Experimental investigations of wakes behind three-
dimensional bodies have been carried out recently by Lemberg (1973), Hansen
and Cermak (1975), Castro and Robins (1977), Robins and Castro (1977), and
Woo e£ al_. (1976), among others.
     The finding of these studies are summarized in Table 1.1.  This table
includes only selected recent results.  In wake flows constraints on the
exponents of the x-dependence of the velocity deficit and length scale can  be
derived from the equation of motion and from a control volume analysis, if  a
self-preserving solution is assumed.  Control volume analyses similar to the
one in Appendix A can be performed for two-dimensional and three-dimensional
objects mounted on a surface or in free flow.  If we define a and b as the
exponents in:  U_av (x) « x~a and fc (x) « x , the control  volume analyses
                inaX
will show that a * b for wakes of two-dimensional objects in free flow,
a = 2b for wakes of three-dimensional objects in free flow and two-dimensional
objects on a surface, and a = 3b for wakes of three-dimensional objects on  a
surface.  The effects of shear have been ignored for the surface flow.  An
analysis of the equation of motion, similar to that in Appendix B, using an
eddy viscosity that varies as x~c, results in c s l-2b.  For a constant eddy
viscosity, c is equal to zero, and thus b = 1/2.   In this report, the eddy
viscosity was assumed to be proportional to the length scale and the maximum
velocity deficit, or c = a-b.  Most of the entries in Table 1.1 satisfy these
constraints with the only glaring exception being the surface-mounted, three-
dimensional object study by Smith, et_al_. (1977).
     There is an additional complexity in wake flows not yet mentioned.

-------
    to
    o
    LU
 Q T>
 •z. ca
 < o
    _
I— «C
»-i Z
o o
O HI
_l CO
O O
     I
CO LU
•Z, LU
o a:
z  a
H-1  2^

If
LU  O
J—  3
LU  H-
O
LU  03
a:  LU
LU  ^
a.  <
X  3
LU
    Lt-
Q  O

^
TJ
3
^J
CO

CO
M-
o
c
o
•H-
Q.
•^~
s_
U
to
CO
o







i- CO
O U
4*3 c
(O CO
O) i.
•r- CO
4<-) H—
tO QJ
co tt:
c s-
HH O


CM
•
S

•> >J
CM (Q CO
(J -M
> CO 10
O CC
CM

to to
"O CO >v
i— to (3 co
O V) U •!->
£•» ^tf ^^ tj
>> S- O Oi
CO -H>
CC CO


.c
+J CO
CTi—
C (O
CO U
_J CO
^
•f- •!- >,
O U 03 CO
O -I- O •!->
i— «4- CO (O
co co o a:
> 0
i-
ta S
CO O
.C i—
CO U.
en
C i-
•r- O
co -o > o
(j CO -H» O i—
 u E •«-
M- C CO

3 O -Q -O i-
co 3E; o co o
X O
CO CO
r— ^
3 a
s-
H-



t.
 (O
0 r-
•1- (/) . 	 s *-^ 	 	 +J
to •»-> r— O CO
c u r^ co c
co co a> o> t- c
E i-S •— f— CO (0
Q O to -t-
1 +•> -M -O C
§c c i— a
3 3 O 0
1— -i- ^r- tri tj»

^•^
^^
1
X i—

O X



1
X r-
— 1
O X







CM CM

XXX


CM
1 III
X XXX

^



X
X











X XX

X XXX












o" f—
en .— .

^^^ f*^ C
•*"•* CT* fO
en r- E
r*~ CT *— ' a.

r— i— .C
«-^ •!-> C CO

o o x: oa
-M r— C (O
to x: 3 >>
ia o o i.
U CO C_» <

CM CM
>x. CO
CO • r—
1 1 1
XXX




CO CO CM
CO r- CO
1 1 1
XXX







CO CM CM CO <* CM ^>

X X X X X X X


GO
CO CM CM CO <• LO ^~
1 1 1 1 1 1 1
X X X X X X X




X
X X



XXX








XX X

X X X X X









X X


*-** *(o| aT
""*" 1 ""*
CD C0| t—
To ^* o "^~
C 0 +•> CO
O 3 c *~~- 
•r- . 3 CO r—
t/) r"* * ^"""i ^C F*^ *- -•
C to (O < — • *~+ U> CT>
CO +J tj- O f^ 08 i— >i
E o 4-^ Is** co en **•"** *o
•r- co co en en r— co 3
Q'f-s i— r— •— ' Cn C7) 4J
1 ^ •» *»— ** *•— *• T5 ^ to
CO O -C CO T- CO
CO 4->4-><4->NlL..O
1. -i- C C C J* E f-
x: E 3 3 ••- in 
-------
Townsend (1976) hypothesized that a wake flow becomes self-preserving (i.e.,
one length and one velocity scale will  be adequate to describe the velocity,
stresses and higher turbulence moments)  because the turbulence attains an
equilibrium state which depends only on  the type of flow and momentum integral,
or, in the present case, the couple.
     However, Bevilaqua and Lykoudis (1978) have shown that wake strength
depends not only on the drag (momentum integral) but also on the structure of
the dominant eddies.  The flow behind a  block-shaped vehicle will  shed vortices
into the wake like the sphere Bevilaqua  and Lykoudis studied.  The resulting
"memory" effect implies that in a block  vehicle wake the turbulent kinetic
energy terms (u'2, v'2, w'2) will  not be self-preserving in the Townsend sense.

-------
                                 SECTION 2

                           SUMMARY AND CONCLUSIONS

     An experimental study of the wake of a moving vehicle was made using a
specially-constructed wind tunnel with a moving floor.  A block-shaped model
vehicle was fixed in position over the test-section floor while the floor
moved at the freestream air speed to produce a uniform, shear-free, approach
flow.  This simulates an automobile traveling along a straight highway under
calm atmospheric conditions.
     Vertical and lateral profiles of mean and fluctuating velocities and
Reynolds stresses in the wake of the vehicle were obtained using a hot-film
anemomenter with an X-probe.  Profiles were taken at distances of 10 to 80
model heights downwind of the vehicle.
     A momentum wake was observed for the block-shaped vehicle.  The wake does
not have a simple self-preserving form.  However, it is possible to collapse
the velocity deficit with one length growth rate and one velocity scale, and
the turbulence with different length growth rate and velocity scales.
     The velocity deficit behind the vehicle was found to decay as (x/h)  '
and the length scale to grow as (x/h)   , confirming the predictions of EH.
                                                                -1 2
However, the turbulent kinetic energy components decay as  (x/h)~ '  and the
turbulent length scale grows as (x/h)0'4 instead of the self preserving solu-
tions of (x/h)~3/2 and (x/h)1/4 which were predicted.
     Velocity deficit solutions are found to the equations of motion for three
cases:  first, allowing scale-lengths to differ for the lateral and axial
directions with constant viscosity (two scale lengths) second, allowing vis-
cosity to vary continuously in the vertical, and finally, a matched solution
where viscosity varied linearly near the surface and then was held constant
above a given height.
     None of the above solutions matched the data in all respects, but the
two-scale-length, variable-viscosity solution was the best.

-------
                                                   -1 2
     The turbulence decayed longitudinally as (x/h)    .   To describe the be-
havior in the lateral and vertical  direction a two-dimensional  fit of the data
was found by least-squares-fit using "orthogonal" polynomials.   Appropriate
constants were determined from the data which allow a complete description
of the wake turbulence.
     Numerical models which predict pollutant concentrations along roadways
can be enhanced by including the physical properties of wakes found in this
study.

-------
                                SECTION 3

                                APPARATUS
3.1  WIND TUNNEL
     A wind tunnel was constructed especially for this study (Fig 3.1).   A
commercially available conveyor-belt assembly was used for the test-section
floor and supporting framework for the entrance, test and exit sections  of the
wind tunnel.  The test section was 4.75 m long (4.15 m of which was moving
floor) with a cross section 0.82 m wide and 0.71 m high.
     The entrance section had the same cross-section dimensions as the test
section.  A bellmouth (diameter of 0.22 m) around the perimeter of the entrance
                                            TM
directed room air into a section of Verticel   (Verticel Co., Englewood  Co.:
paper triangular-cell honeycomb; cell length = 0.15 m and hydraulic diameter
= 0.01 m).  Two fiberglas screens (16 x 18 mesh) were placed downwind of the
honeycomb to further lower the freestream turbulence.
     The moving portion of the test-section floor was the rubber conveyor belt
(0.75 m wide).  A notched overlapped joint and large end rollers produced a
smoothly running belt.  Rails along the tops of the walls were installed to
support an instrument-traversing mechanism.  The longitudinal position of a
probe was set manually; vertical and lateral positions were remotely set and
read from an operator's desk.
     The exit section contained a section of Verticel and a lateral con-
traction to a 0.71 m wide by 0.71 m high cross section.  A cast aluminum fan
with three blades (diameter of 0.61 m) was powered by a two-speed (1140/1725
RPM), 250 W electric motor (115 V, 60 Hz).  The fan was mounted in a metal
"venturi panel" (i.e., shroud) for efficient operation.  Baffle plates of
different porosities were inserted into a slot just upwind of the fan to fine
tune the air speed or to obtain a variety of air speeds for velocity probe
calibrations.  The maximum air speed, obtained with no baffle plate and  the
fan motor on high speed, was 4.0 m/s.

-------
     The operating speed of the tunnel  during all  experiments was  1.9 m/s;
the turbulence intensity (root mean square of the  velocity fluctuation divided
by the mean speed) was less than 1%.
     In performing additional  tests to  determine possible Reynolds number
effects and the extent of the blockage  effects resulting from the  size of the
models, two additional wind tunnels of  a more conventional nature  were used:
the Fluid Modeling Facility's Meteorological  Wind  Tunnel (Snyder,  1979), and
the Air Pollution Training Wind Tunnel  with test sections of 3.7 m wide x 2.1 m
high x 38 m long and 1 m wide x 1 m high x 3 m long, respectively.

3.2  VELOCITY MEASUREMENTS

     Velocities were measured with Thermo-Systems .(St. Paul, Mn.)  hot-film
anemometers.  X-configured probes (model 1243-20)  were used to obtain trans-
verse velocity components as well as the streamwise component.  The bridge
outputs of the anemometers (model 1054-A) were suppressed with signal condi-
tioners (model 1057), attenuated, and sent to the laboratory's Digital Equip-
ment Corporation POP 11/40 minicomputer.  Software in the computer controlled
the analog to digital conversion or sampling rate, linearized the  signal, and
provided real-time displays of results.  The probes were calibrated in posi-
tion on the instrument traversing mechanism against a pitot tube located
nearby for determining mean test section velocities obtained with  the various
baffle plates.  The X-probe was oriented to measure either the streamwise and
lateral components or the streamwise and vertical  components.  Calculated
values were mean streamwise speed, fluctuating streamwise and transverse com-
ponents, and the associated Reynolds stress.  Most measurements were made at
100 samples per second for 30 seconds.   A few ten-minute samples at 500 samples
per second were taken at 30 heights downwind of the vehicle at various vertical
positions in order to calculate turbulence spectra.  From these, an eddy visco-
sity formulation was derived for use in a theory presented below.

3.3  MODEL VEHICLES

     Simple block-shaped models were constructed of wood; they had proportions
of height, width and length of a typical automobile (Fig 3.2).  Each model
                                        8

-------
was mounted on a baseplate with wheels and rubber tires.  Approximate scale
ratios of 1/32 and 1/8 were used.
     The 1/32-scale block-shaped vehicle was patterned on the dimensions of
an intermediate size American automobile:  height = .043 m, width = .055 m,
length = .145 m, ground clearance = .009 m, and wheel  diameter = .024 m.  The
1/8-scale vehicle was four times as large in all respects.

3.4  EMPTY WIND TUNNEL FLOW CHARACTERISTICS

     Homogeneity of the mean flow was checked with the tunnel operating at test
condition by measuring vertical profiles at various positions in the test
section.  Fig. 3.3 shows profiles on the centerline for distances of 1, 2, 3
and 4 m from the front of the test section.  Fig. 3.4  displays the lateral
homogeneity with profiles at a distance of 1 m from the front of the test
section at lateral positions of -.205, 0, and .205 m.   The air speed was 1.9
ms  ; there was little scatter down to the belt surface.
     Vertical profiles of the turbulence intensity at  a distance of 1  m from
the front of the test section and lateral positions of ±.205 m are shown in
Fig. 3.5.  The turbulence intensity is seen to be constant and below 1% except
very near the belt, where probe sensitivity near the surface is suspect.

3.5  WIND TUNNEL BLOCKAGE EFFECTS AND REYNOLDS NUMBER  INDEPENDENCE

     In order to determine if the velocity and turbulence profiles for the
large vehicle were affected by blockage in the moving-belt tunnel, several
tests were performed with both the small and large vehicles.  These experi-
ments were performed in the moving belt tunnel and in  two other wind tunnels
so that various values of the base pressure coefficient,

                  P   - P
                  rsy.    rb
could be determined as a function of Reynolds number and blockage ratio.   In
(3.1) PS  is the free-stream static (reference)  pressure upwind of the vehicle

-------
(F1g. 3.6), P^ 1s the static pressure pleasured on the rear surface, and 1/2
pU^  1s the reference dynamic pressure.
     Now (3.1) can be written as
                                               P  - P
                                               '&    •>
                                               P  - P
                                               '*    rs
     or
                                          f>0 - f»
                                                                       (3.2)
where P  is the local  static pressure and P  is the total  pressure given by
P_ = 1/2
             + Ps .   The pressure ratios in (3.2)  can be easily determined
from measurements.
     In Fig. 3.7 the spanwise variation in C  for various blockages is shown.
While the pressure is not constant across the rear as one might expect, the
variation is small, and is most likely due to asymmetries in the vehicles and
the flow not being exactly normal  to the front of the vehicles.  Since the
pressure variations were small, the pressure at the tap at y/W = -0.25 was
used in all calculations.
     In Fig. 3.8 data are presented showing that while -C_  is a function of
                                                         pb
blockage ratio it does not appear to be a function of Reynolds number.  However,
there was considerable scatter in the data.
     The pressure coefficient -C   is plotted against blockage ratio in Fig. 3.9.
                                pb
For the experiments performed in the non-moving belt tunnels, the vehicles were
placed near the tunnel entrance immediately downwind of the contraction.  This
insured a small floor boundary layer (thickness fi).  The boundary layer was, of
course, relatively more important for the small vehicle than for the large
vehicle.  This may be the reason for the sudden change at small blockage ratios
which otherwise is not reasonable.
     The drag coefficient, Cn, can be estimated by assuming that the normalized
                                                          2
integrated pressure coefficient on the front face is (0.8) , as reported by
                                       10

-------
Mason and Beebe  (1978), to give
                 =  (0.8)%(-CP\
                                                                      (3.3)
     Table  3.1  shows the calculated  drag coefficient as  a  function of blockage
where A^ and Ay are the cross-sectional areas of the model and wind tunnel,
respectively.
             TABLE 3.1.  DRAG COEFFICIENT AS FUNCTION OF BLOCKAGE
                        (% blockage)

                         .51
                        3.84
                        6.50
                                             0.775
                                             0.876
                                             0.957
We see that  the drag coefficient  is about 20% greater for  the large car in the
moving belt  tunnel than for the small car in the same tunnel (Table 3.1).
     An empirical formula (Castro and Fackrell, 1978) for  estimating the
blockage effect on drag has the form
           CD   =CD(l-AM/AT)m
                                                                      (3.4)
where
       p
          is  the observed drag coefficient.  For the block  vehicle, m was
found to be  3.31.  For two-dimensional fence flows, Castro and Fackrell found
m varied between 2.6 and 7.5 depending on the ratio 
-------
C, acting on the vehicle to a surface integral  of the velocity deficit,
u(x,y,z), assuming that an integral  of the surface pressure  is small.  That
is:
                    CO  CO
                           ,1 T^ J.. J,
                                                                         (3.5)
     A control  volume analysis  of the conservation  of momentum yields  a  relation
ship for the drag, D, on the vehicle:

                                           x
                                                                         0.6)
                   0  "00             -CO -00

where T  (x,y,o) is a  surface shear stress defined in Appendix A.
       i A
     A centerline vertical  profile of u/U and a lateral  profile of u/U at
z/H = 0.5 are plotted  in Figs.  3.10 and 3.11, respectively,  for the small  and
large vehicle at x/H = 15.   These two figures imply that the couple computed
from (3.5), scaled by  vehicle size, would be approximately equal  for both
vehicles.  Similarly,  the first term on the right side of (3.6) would be the
same for both vehicles.   The previously discussed difference in drag coefficients
for the two vehicles can be explained as a difference in the integrated surface
shear, the second term on the right side of (3.6).
     Reynolds -number independence of the vehicle wakes is further supported by
a comparison of velocity fluctuations at x/H = 15 shown  in Fig. 3.12.  Thus,
similar wakes exist for the two sizes of vehicles independent of the somewhat
larger coefficient of  drag found for the large vehicle.
     In summary, wind  tunnel blockage is not a problem and the wake flows are
independent of Reynolds number.
                                      12

-------
                                  SECTION 4

                WAKE THEORY AND MEASUREMENT OF VEHICLE WAKE

4.1  REVIEW OF VEHICLE WAKE THEORY

     A car moving at speed U through a calm atmosphere (windspeed  <1 ms   )  is
equivalent to a car held fixed in space over the roadway with  the  road and
air moving past the car at speed U.   The coordinate  system  used  here has  its
origin at the rear of the vehicle.  Several heights  downwind of  the vehicle,
beyond the recirculation region, the velocity deviations (u,v,w  « U) from  the
undisturbed flow are small enough that a perturbation  analysis of  the equations
of motion can be carried out (details are found in EH).   The first equation of
motion has the following linear form
where it is assumed that
                                                                        (4'2)
and the eddy viscosity can be estimated  as

               (X)  GC,<(xl IAWWC(X),                              (4.3)
where £(x) is a vertical  scale length in the  wake  and  um.,  (x)  is  the maximum
                                                      max
velocity deficit in the wake.
     In this (EH) theory it is implicit that  the vertical  and  lateral length
scales are Identical.   The pressure gradient  term  in  (4.1)  can  be neglected;
                                     13

-------
the solution to the resulting equation is
                                                                        (4.4)
where
        Y[  =
                                                                        (4.5)
Y and A are constants, and H is the height of the vehicle.   The  value of A  can
be estimated by approximating the couple acting on the vehicle by the drag  on
the vehicle times one half the height of the vehicle.   This  results  in (see EH)
                   we.
                                                                        (4.6)
     The turbulence is estimated by assuming it has  a self-preserving form and
that the turbulent kinetic energy is proportional  to the eddy shear stresses  in
the wake.  This yields the results:
                                             rs/a
                                                                        (4.7)
                                                                        (4.8)
                                      14

-------
     This theory is reasonable for the velocity field except near the surface,
where it is clear that a much stronger shear layer exists than is possible with
a constant eddy viscosity.  Also, the theory does not allow for obstacles that
are wider than they are high.  Fig. 4.1 shows a centerline velocity profile
taken at x/H = 60 compared with the original theory.   It is immediately obvious
that the wake contains a very strong shear layer near the surface that is in-
compatible with a constant eddy viscosity in the vertical direction as proposed
in (4.3).
4.2  VERIFICATION OF THE EH THEORY

     One of the important predictions of the theory in EH is that the velocity
deficit and lateral length scale vary as
                           
-------
where
     Now (4.10) can be written as

                                                                        (4.11)
The left hand side of (4.11) was computed from the data on a point by point
basis using cubic spline fits to velocity deficit profiles to obtain |^.   Near
                                                                     oZ
the top of the wake and near the maximum velocity deficit, u'w1  and |^- become
                                                                    oZ
small so that errors in u'w'/dr) can be large.  However,  the ratio was found
                              dZ
to be nearly constant over the range between these extremes and  an average value
was computed for each downwind distance.  These are plotted in Fig. 4.6 which
shows that the left hand side of (4.11)  can be reasonably  fitted with a line
of slope -1/2.  Since we have already shown that u _•   a (n-)"   » this implies
                                                  max    rt
again that

                         '4                                          (4.12)

     However, the theory developed in EH does not describe the lateral spread
of the wake as well as desired.  The EH theory assumes that £(x) is the correct
length scale in both the vertical and lateral directions.   This  is equivalent
to assuming the vehicle is as wide as it is high which is  incorrect.  The line
through FV/^) in Fig. 4.6 lies well  above the line through u^w7/^)  imply-
ing different length scales for the vertical and lateral directions.

4.3  MODIFIED WAKE THEORY

     A modified theory which accounts for different scale  lengths in the
lateral and vertical directions is derived in Appendix B.   The solution for
                                      16

-------
the velocity deficit is the same as (4.4) except that now

                                                                       (4.,3,
and the constant A in (4.4) is given by

                                    1/4

                                                                        (4.14)
C
where A and y are constants with values of 1.14 and 0.095, respectively.   The
length scales are
4.4  HEIGHT-DEPENDENT-VISCOSITY WAKE THEORY

     A further modification to the theory was made in an attempt to improve the
agreement between the theory and data for the region of strong shear near the
surface.  The eddy viscosity was allowed to vary in the z direction as described
in Appendix C, where the details of how the height dependence of the eddy
viscosity was determined from spectra of w'2 are also presented.  A numerical
solution was then found.
     Appendix D contains a simplified approach to a height-dependent eddy
viscosity solution using a linear approximation in the strong shear layer and
dividing the solution of the equation into two regions.  The lower region has a
linear increase in eddy viscosity and the upper region has a constant eddy
viscosity.  Results from this technique did not differ substantially from the
numerical solution using a continuously variable eddy viscosity and are there-
fore not presented.
                                      17

-------
     The three theories, original  EH,  modified lateral  length scale,  and
height-dependent-viscosity, are compared with vertical  and  lateral  velocity
deficit profiles in Fig. 4.7 and 4.8,  respectively.   The height-dependent
eddy viscosity theory lowers the height of the maximum  deficit by a factor
of nearly two and increases the shear  near the surface  (Fig.  4.7).  However,
the maximum in the data occurs still  lower.  The improvement  in describing the
lateral extent of the wake by the modified lateral  length scale theory is
evident in Fig. 4.8.
     Lateral profiles of velocity deficit at a constant similarity height, c,
are shown in Figure 4.9.  These profiles are reduced to similarity coordinates
and compared with the height-dependent-viscosity theory in  Fig. 4.10.  Vertical
profiles of velocity deficit are compared with the  height-dependent-viscosity
theory in Fig. 4.11.

4.5  TURBULENT KINETIC ENERGY TERMS

     Preliminary analysis of the fluctuating velocity data  indicated  that the
lateral length scale determined for the velocity deficit would not collapse the
profiles of the turbulent kinetic energy components  (TKEC).  The decay rate of
the TKEC was also found to be different than that proposed  in EH where complete
self-preserving flow was assumed for the wake.
     The turbulent energy budget equation can be written (Tennekes and Lumley,
1972)
        _L IJ.  ^_
        Z ui
                                                                        (4.16)
where
                                                                        (4.17)
                                    18

-------
and the convention of repeated  indices to  indicate summation is used.
     Assuming as for the velocity deficit  solution that longitudinal  changes
scale with x, while crosswind changes scale with I1 where a'/x « 1,  let
li2 = 1/3 u.u. and make order of magnitude  estimates of the terms in (4.16).
Assuming there is only one  velocity and length scale in the wake:
           j-fij  (* ^ V Uo
-------
                                                                       (4.19)
     A transverse Eulerian integral  length scale defined  by

           ^vjjjv^     j                                   (4-20)
                 o    V'(o)
was computed from data obtained using two x-film probes.   This  length  scale
is believed (Tennekes and Lumley,  1972)  to be comparable  to  the Lagrangian
integral length scale for most  turbulent flows and  is  thus a measure of the
size of the dominant eddies within the wake.   Thus,  I1  should be related to
£r; if we empirically determine the x-dependence of  i^ and assume I1 « £E
we can evaluate a and 6.
     The two probes  were calibrated and  then  positioned in the  wake of the
block vehicle at one half of the vehicle height. One  probe  was positioned on
the wake centerline  while the other was  moveable in  the lateral direction.
Profiles of v'(o) v'(y) were obtained at downwind distances  of x/H =  15,  30,
45, and 60 (Fig. 4.12).  For the greater downwind distances, v'2  in the wake
was not large compared with background values,  so vh'ackaround was subtracted
from v'2(0) for each profile.
     The length scale ££ was determined for each of the profiles  by a trapezoi-
dal integration.  A power-law fit corresponding to a = 0.40  is shown  in Fig.
4.13.  The value of $ computed from (4.19)  is therefore 1.2.  Using the computed
length scale £r to normalize the space correlations of Fig.  1 collapses the
data nicely (Fig. 4.14).

     The maximum values of u'2 and v'2 were determined from  the vertical  center-
line profiles obtained at various downwind  distances.  These are shown in
                                    20

-------
Fig. 4,15 along with the least squared error fit which gives p *  1.2,  in
agreement with the Independent estimate from (4,19).
     Thus, the vehicle-generated turbulence in the wake was  found to decay as
(x/H)"1'2 and the lateral length scale was found to grow as  (x/H)0'4.
     The spatial variation of the turbulent kinetic energy is assumed  to  be
given by
                                                                       (4.22)


                                          u>*2/£H

and

                          =  1.0,                                      (4.24)
     At (x/H) = 30 detailed measurements  were taken  in  the  lateral  direction
using a 3-min. averaging period.   The function F  (x,o>)  was  determined  by  a
                                               w
least-squares fit using orthogonal  polynomials of the fourth  degree (Clark,
Kubik and Phillips, 1963):

          Fc(x,u) = ₯00 + W^f01*  u(₯02 +  u (>F03  + u
                        + X4 (*40 + ^(^  + w(V42   +  u()  = 0.0  when:
 c                       c
            i)  |xj 1  0.55 or  | w | >_  0.64
           1i) x ±0-0   and tt >1.82X  + 1.15
          111) x > 0.0   and u >-1.82x  + 1.15
                                     21

-------
The coefficients 1n (4.251 are found 1n Table 4.1.   From these data the  con-
stants 1n (4.211 were found to be a^ = 0.048, a2 ~  0.040, and  a3 =  0.030.
     In Figs. 4.16 to 4.18 normalized cross-sections of the turbulence com-
ponents u'2, and w'2 are shown fitted by the orthogonal  polynomials of (4.25).
The contours of FC are presented in Fig. 4.19.
     Using (4.21J with the numerical fit from Table 4.1, vertical and lateral
profiles of the fluctuation velocity components  can be estimated.   Using simi-
larity coordinates, data at various longitudinal distances can be plotted and
compared to these estimates. F1gs. 4.20 to 4.22  show linear and similarity
plots of vertical profiles of u'2, v'2 and w'2,  respectively.   Figs. 4.23 and
4.24 show linear and similarity plots of lateral profiles u'2  and w'2.

4.6  REYNOLDS STRESSES IN THE WAKE

     The wake theories presented in this paper and  that of EH  make  no predictions
regarding the decay of the Reynolds stresses. Bevilaqua and Lykoudis (1978)
define a hierarchy of self-preserving wakes starting with order one, where mean
velocity profiles are self-preserving; order two, where in addition Reynolds
stresses are also self-preserving and so on.
     From the data for center!ine variation of the  stresses (Fig. 4.25a  for
u'w'), it is easy to show that the stresses decay as (x/H)  '   and  hence using
                      25
the EH value of (x/H)    as the length scale growth rate, the  data  collapse as
shown in Fig. 4.25b.  Fig. 4.25c is the same data plotted using TKEC similarity
coordinates.  It is easily seen that the wake is self-preserving of order two
in terms of EH similarity variables.  Fig. 4.26a and b show lateral profiles of
u'w1 at various (x/H) distances, and then the same  data in EH  similarity co-
ordinates.  The uV stresses, shown, are similar to the u'w1  stresses.
                                       22

-------
TABLE 4.1   COEFFICIENTS OF POLYNOMIAL TWO-DIMENSIONAL  FIT TO  TURBULENT
           KINETIC ENERGY FUNCTION F .
                              ^00 =  0.3163277 x 10"1
                              T01 =  0.1130908 x 102
                              T02 = -0.4320938 x 102
                              ^03 =  0.6065336 x 102
                              ₯04 = -0.3218438 x 102
                              f20 = -0.1703226
                              Y21 = -0.8419376 x 10
                              ₯22 = -0.1640925 x 103
                              Y23 =  0.5061181 x 103
                              ^24 = -0.3599435 x 103
                              *40 =  0.2386405
                              *41 * -0.8499160 x 102
                              *42 =  0.9322793 x 103
                              ^43 = -0.2115435 x 104
                              744 =  0.1360754 x 104
                                    23

-------
                               REFERENCES
Abramowitz, M. ,  and L.A.  Stegun, 1964:   Handbook of Mathematical  Functions.
NBS Appl. Math. Ser., No.  55, 1046 pp.

Arya, S.P.S. and  M.S. Shipman, 1979:   A model  study of boundary layer flow and
diffusion over a  ridge.  Proc. AMS Fourth Symp.  on Turbulence, Diffusion and
Air Pollution, Jan. 15-18, Reno, NV.  Am. Meteorolooical  Soc.,  Boston, MA,
p 584-91.

Bevilaqua, P.M. and P.S. Lykoudis, 1978:  Turbulence memory in self preserving
wakes.  J_. Fluid  Mech., 89, 589-606.

Castro, I.P., 1979:  Relaxing wakes behind surface-mounted obstacles in rough
wall boundary layers.  J_.  Fluid Mech..  9_3, 631-659.

Castro, I.P. and  J.E. Fackrell, 1978:  A note on two-dimensional fence flows,
with emphasis on  wall constraint.  J_. Indust.  Aero., _3,  1-20.

Castro, I.P. and  A.G. Robins, 1977:  The flow around a surface-mounted cube
in uniform and turbulent streams.  J_. Fluid Mech., 79, 307-335.

Clark, R.E., R.N. Kubik, and L.P. Phillips, 1963:  Orthogonal  polynomial least
squares surface fit.  Comm. ACM., 6_,  146-148.

Counihan, J., J.C.R. Hunt and P.S. Jackson, 1974:  Wakes behind two-dimensional
surface obstacles in turbulent boundary layers.  J_. Fluid Mech., 64, 526-563.

Counihan, J., 1971:  An experimental  investigation of the wake behind a two-
dimensional block and behind a cube in a simulated boundary layer flow.
P-'I-i-i- Lab Note, RD/LN115/71, Central  Electricity Research Laboratories,
Leatherhead England, 45 pp.

Eskridge, P.E. and J.C.R.  Hunt, 1979:  Highway modeling.  Part I:   Prediction
of velocity and turbulence fields in the wake of vehicles. J_.  Appl. Meteor.,
18., 387-400.

Goldstein, S., 1938:  Modern Developments in Fluid Dynamics.  Oxford University
Press, Oxford.  Reprinted by Dover.

Hansen, A.C. and J.E. Cermak, 1975:  Vortex-containing wakes of surface obstacles,
Project THEMIS Technical report no. 29.   (Fluid Dynamics and Diff. Lab. Rpt.No.
CER75-76ACH-JFC16,) Colorado State University.  Ft. Collins, CO. 163 pp., Dec.

Hinze, J.O., 1975:  Turbulence.  McGraw-Hill, 79C pp.

Hunt, J.C.R., 1971:  A theory for the laminar wake of a two-dimensional body
in a boundary layer.  J. Fluid Mech., 49, 159-178.
                                      24

-------
Hunt, J.C.R., 1974:  Wakes behind buildings.  Aeronautical  Research Council
(London), Paper ARC35601, Atmos. 229,

Hunt, J.C.R., 1980:  Personal communication.

Kreyszig, E., 1962:  Advanced Engineering Mathematics. John Wiley and Sons,
Inc., New York, NY., 856 pp.

Lemberg, R., 1973:  On the wakes behind bluff bodies in a turbulent boundary
layer. Ph.D. thesis, University of Western Ontario, 152 pp.

Mason, W.T. and P.S. Beebe, 1978:  The drag related flow field character-
istics of trucks and buses, In: Aerodynamic Drag Mechanisms of Bluff Bodies
and Road Vehicles, Edited by G. Sovran, Timorel  and W. Mason, Plenum Press,
New York, NY.  pp 380.

Robins, A.G. and I.P. Castro, 1977:  A wind tunnel  investigation of plume
dispersion in the vicinity of a surface mounted  cube.  I.  The flow field.
Atmos. Environ.. !]_, 291-297.

Schlichting, H., 1960:  Boundary Layer Theory, Sixth edition, McGraw-Hill,
New York, NY., 747 pp.

Smith, F.T., R.I. Sykes and P.W.M.  Brighton, 1977:   A  two-dimensional boun-
dary layer encountering a three-dimensional hump.  J_. Fluid  Mech., 83, 163-176.

Snyder, W.H., 1979:  The EPA Meteorological Wind Tunnel - Its Design, Constru-
ction, and Operating Characteristics.  EPA 600/4-79-051, U.S. Environmental
Protection Agency, Research Triangle Park, N.C., September, 78 pp.

Tennekes, H. and J.L. Lumley, 1972:  A First Course in Turbulence, MIT Press,
Cambridge, MA., 300 pp.

Townsend, A.A., 1976:  The Structure of Turbulent Shear Flow, second edition,
Cambridge University Press, Cambridge, England,  429 pp.

Woo, H.G.C., J.A. Peterka and J.E.  Cermak, 1977:  Wind tunnel measurements in
the wakes of structures. NASA Contractor Rpt. NASA CR-2806, Colorado State
University, Fort Collins, CO., 243  pp., March.
                                      25

-------
                                 APPENDIX A

                           CONTROL VOLUME ANALYSIS

     The overturning moment acting on the vehicle can be related to the
velocity deficit in the wake with a control volume analysis.  Consider the
control volume enclosed by the box shown in Fig. A-l .  The vehicle is excluded
from the control volume and the effects of the vehicle on the flow are combined
into the overturning moment, C.  The small areas where the vehicle tires con-
tact the ground are ignored in the analysis.
     The vector equation for conservation of moment of momentum is
where z ft is the sum of externally acting moments on the control  volume, r is
a position vector, q is the velocity vector (U-u,v,w), n is an outwardly
directed unit vector, t, j\ t are the unit vectors in the x,y,z directions
                  ->•   +   •*•       ->•
respectively, and M = r x F where F is a force on the control surface.

     Steady state conditions are assumed, i.e.  ^'  ' =  O .  Since the over-
turning moment of the vehicle is about the y-axis, the dot product of j" with
(A-l) is taken; this, together with an assumption of constant density gives
                                                                        (A-2)
     By definition, a shear stress component is positive when the shear stress
and vector outwardly normal to the surface of action both act in the positive
or negative directions.  The shear stress, TV   , acts in the x.{ direction along
                                            X1xj               J

                                       26

-------
a surface normal to the *. direction.
     The external moment about the y-axis resulting from forces acting on
Face 1 are due to pressure and shear stress:
           -
Combining and integrating over the area of the face yields the total  torque
about the y-axis on

          2, Y,

Face 1:
          o -Y;
In a similar manner the external moments on the other faces are found to be
on
               2rt
                                      27

-------
         2. Y,



Face 2:
         Xi ?,



Face 4:         -Xt   (X| Vl/
        -*, o
Face5:
                    -x, -Y
                                                 <
                                                J   /
                        ^ t 2, T2X U^, 2,)] J
                      Xi Y

Face 6                 0 0

(and vehicle surface):   \\ [^ * *(*y ^/Oj] Ju^x - L
                                 28

-------
     C 1s the overturning moment acting on the vehicle which 1s positive In
the y direction; C 1s the result of all forces from pressure and stress acting
on the surfaces of the vehicle.
     Now the left hand side of (A-2) can be written
                                                                       (A"3)
             x,. r,
                                                                - C  .
We also can write the pressure as the sum of a mean and perturbation
and we assume the perturbation pressure to be zero far upstream of the vehicle,
i.e. p(x-i, y, z) = 0.  The T'S may also be considered perturbation quantities
since TUDstream = ^ (uniform flow).  Equation A-3 can now be written as
* From symmetry, T   (x,Y,,z) =^r   (x, -\J,z), etc.
                                      29

-------
                   2, Y,
                   O -Y|
                     Xt.
                    -"X, o

                 •it
                   ^*< * '{
                                          -c
    Now the moment of momentum crossing each face of the control surface must
be computed, i.e. the right hand side of (A-2).
Face (1)
                     a -
also      / x? v;? \. "f  - zU          and
therefore    I  =  - f     2 U
             '        ^j  j
                      o -X
                                30

-------
Face (2)
also     trx£)-  = 2  U-utvV,,*) - XW(X,-V;,E);
and
                Xl
       I 2 = -
Face (3)     -                   avv(
                                 v(i  f s
also    (rx)«J = 2  iy-u)-Xj.w   ^J      r\.? = U-u . Hewce,

       I3  = f
               o -^
and
also
and    (w«=  V
Thus   T4r f
            -*< o
                            31

-------
Face C5)        ,r*x*
-------
Now  |wl£<|u|  (from EH),  so that  X2Uw«zUu in the first integral.
Let  Z^froo; then p(x,y,Z1 )-*0, w(x,y,Z1 )~»0, andTzx(x»y»Z1 )->0.
Also let Y]-*oo; then T   (x,Y] ,z)—^0 and V(X,YI ,z)-»0.  Let -Xj»-»,
and  since X2 is arbitrary it can  be  replaced with x, reducing  (A-5)  to
                                                                        (A-6)
               ,—           -      -*   j       i  \
              "^                             -CD -CD
The first equation  of motion in perturbation form is
                                                                        (A-7)
Since there are  no momentum sources in the control  volume  from x = x to
x = oo »  y = -eo  to y = oo, z - 0 to z -oot eq.  (A-7)  holds everywhere
and can  be integrated over this volume.
Integrating (A-7) with respect to y from -co to  «o yields
       O                Oo           oo              ct>
                    1  \ 3? Ju 4. J.  V 3?** ,j    1 f 3?n  i
                    f   \3x^^f    jr^n^^V        (A-8)
      -co              "~°°           -oo             ~ce>
                    _
Now    f JIT^     ~   x^5     ~   )   yielding
         -co
        -co            -co         -co
Integrating  (A-9) with  respect to x from x to CD yields
                                                    00 OO
         X -co                  X -oo               X "00
                                    33

-------
or
 '     CO           CO      °0 00

                     ••    "*"»'••
      —CD          —CO      X -°°



Multiply by  z and Integrate with respect to z from 0 to co


      OO GO            OO OO          OO OO 00
      0 -«>            o-oo         O X -c»
      o   x -oo              -oo *  o



                           ,00   n
so that (A-12) becomes
      C=
           o -oo            ^            "0° "c°
                           - CD
           0-00                         -0) K 0

            OO
             i  ixP^i
            -CD C»


                               34
       o-oo           0 -»          OM*
And (A-6) can now be written as
          oo OQ             °o            co  x

-------
Since
        5°                *
        \T   J  -   V  'i— 
Now (A-7) can be written for distances sufficiently far downwind that the
perturbation pressure is negligible (see section 4.1), as
                                  .  £i "I
                                  +        J
                               J

Multiply this by  z, integrate the right hand side by parts from  z  = 0 to •
and then integrating with respect to y from - oo to cx>, to get
          CD GO
or
        -00 o
    The couple on the vehicle for a given situation is a value that could be
determined by measurement.  At distances sufficiently downwind for (A-19) to
hold, we can therefore conclude that the second integral on the right of  (A-17)
is a constant, not a function of x.  It is assumed that the second integral in
(A-17) 1s small and therefore the couple on the vehicle determines the wake
strength.  Hunt (1974) indicates that the second integral can be neglected
for three dimensional objects.

                                        35

-------
     When the stress is  approximated  by  a  viscosity  that  varies in the vertical
direction the analysis is similar to  that  given with the  following modifications
Equation (4.1) in section 4.1  takes the  form
where H*(x,z) is the vertical dependence of viscosity
Multiplying by z and integrating with respect to y and z yields
           oo
                            y           I i_ J     '  ' 13    I a U
          -a, o                        jo       J*   J   J*

(A-21) then gives an integral  constraint on the definition of H*(x,z) i.e.
        CO
                                                                              (A-22)
 since the right hand side of  (A-21) must be zero.   From  (A-21)  and  (A-22)
        oo
           0             y                                                    (A-23)
The analysis of the couple would be the same since (A-16) would take the form
        oo                   00
                                                                              (A-24)
In the work presented in this paper it is assumed that (A-22) is satisfied.
                                       36

-------
                                 APPENDIX B

                           MULTI -SCALE WAKE THEORY

     The following derivation follows very closely the form in EH.  The
assumptions are the same as those given in Section 4.1 and we use the same
notation.
     The linearized first equation of motion is given
where the shearing stresses are assumed to have the form
where W and H are the width and height of the vehicle, respectively, and
We have allowed for different length scales in the lateral and vertical and
assume them to be related by
As in EH it can be shown that 3p/3x can be neglected.  (B-l) now becomes

                                           U/1  1\  1                    (B-5)
                                       37

-------
Assume  a solution of the form
where A  Is a constant to be determined and f <. 1.
Now, let
            = 2/jfut)    
-------
 From the control volume analysis given in Appendix A the overturning couple on
 the vehicle is given by
              C

      C*r(
              O
       CO  CO

                       " ~~               *                      (B-12)
             -co o
The approximation on the  right  is obtained by taking the overturning moment to
be the drag force times one-half the vehicle height.
     If we substitute (B-6)  into (B-12) the integral constraint becomes
                                                                      (B-13)
                                  -CO O
Hence,
C
                H                »   or
From (B-ll) and (B-14)
              x3 x"3m  x"m = constant,                                 /B_15v
                                                    -3/4
which implies  that m = 3/4.  Thus, we have %x(x)<< x     and
J?(x)flCx1/4.   If we define f(x) = ^AH (x/H)1/4,  and  note that
     = AU (x/H)'3/4,  (B-8) becomes
  max
                                   39

-------
describing the behavior of the velocity deficit in all planes normal to the
wake.
     The solution to (B-16) is known (EH) to be
         A   ^ / -i	\'/z
where   JU, ~
     The constant A can now be computed using (B-12) and (B-17);
                                          co oo
                                                                          (B-is)
                                                                 j
yielding
                ^    X^         --^  f /Vl
                                                                          (B-19)
                                      40

-------
     The solution for the velocity deficit Is now known (B-6, B-17, and B-19)
except for two constants y and \.
     To determine X we use (B-2);
                                                                          (B-20)

and

           ^  T*,  "V-r (X) =  \7\i~> / ( ^ \
                                                                          (B-21)
     The right hand sides of (B-20) and (B-21) have been determined at various
downwind distances by fitting vertical and lateral velocity profiles with cubic
splines which were differentiated to determine the velocity gradients (Fig. B-l),
Above the maximum velocity deficit the velocity gradient was well behaved and
averaged values of -u'w'/(3u/3z) and -u'v'/(3v/3z) are plotted in Fig. 4.5.
The straight lines fitting these points are least squares linear fits to the
data.
     The value of A. can now be determined by using, say, x/H = 10 and the
fitted values to the profiles yielding
                                                                         (B.22)
     To determine y we rewrite B-3 as
                                                                         (B~23)
                                      41

-------
or
       l/V      (*/H)

                                                                (B-24)
At x/H = 10, vT2/U2H2 = (0.007)2 and by use of (B-19) one finds  that
          - 0,095".
                                                                (B-25)
                                    42

-------
                                 APPENDIX C

                   HEIGHT-DEPENDENT-VISCOSITY WAKE THEORY

     Assume the perturbation equation of motion (B-1)  can be written as
where H*(x,z) is some function of height having a continuous derivative.   From
Appendix B we can express (C-l) and its solution as
(C-2)
                                    ^W^l
                                   ,*)^)* —-J
and

                                                                         (C-3)
where

                                                                         (c'4)
                                      43

-------
      Substituting  (C-3)  Into  (C-2) yields  for  f  the  partial  differential
          'Xi                                    <\i
 equation  (H(0  is  the  transform of H*(x,z) and H(c)  <1)
                                                                         (C-5)
Equation  (C-5) can be solved in part by assuming it is separable,  i.e.
This yields                                                              ^C"6^
and
                                                                         (C-7)
where  (C-7)  is subject to the boundary conditions
                                                                         (C-8)
     Analytic  solutions are easily found for  (C-7) and  (C-8) only when H(c)  is
 a  simple  function.  Hence, (C-7) and  (C-8) have been solved numerically using
 the  Runge-Kutta-Nystrom method  (Kreyszig, 1962).
                                    'X/
     To arrive at an expression for H(c) we have taken  data along the center-
 line (y=0)  at  x/H = 30 for z/H  = 0.15,  0.29,  0.44, 0.59,  0.74,  0.88, 1.03,  1.18,
                                       44

-------
1.32, 1.47, 1.76, 2.06, 2.35, and 2.94, which allowed us to compute spectra
at each height (see Fig. C-l).  We further assumed that
where (w12) '   is the standard deviation of the vertical velocity, U(c) is the
mean wind speed at height c, and N is the frequency containing the maximum
energy.  A best N was determined for each height from the spectra together with
estimated upper and lower bounds.  The product (w'2) '   U (?) N is normalized
by dividing by its maximum value.  These results together with error ranges
are plotted in Fig. C-2.  The following function describing H(e) was found by a
least-squares fit to the data in Fig. C-2, after changing variables from z/H
to c.
                                                                        (C-11)
                                 3 -0,
     In order to describe the vertical  change in viscosity we plotted

u'w'/(3u/3z) from data taken with the 1/8 scale block vehicle.   The vertical
velocity gradient was estimated at each point with a cubic spline fit.   The
estimate naturally contains error and is partly responsible for the scatter in
Fig. C-3.  Careful study of Figure C-2  and C-3 shows that, except for scatter,
the trends in both plots are the same and the maximum occurs near z/H =  1.25
in both plots.

                                     45

-------
                                  APPENDIX D

               SIMPLIFIED HEIGHT-DEPENDENT-VISCOSITY WAKE THEORY

     A simple approximation for the height-dependence of the eddy viscosity can
be made that will lead to an analytic solution.  If we assume that viscosity
                      ^
distribution function H (c) has the form

then (C-8) can be solved as follows.  Substituting (D-l) and its derivative
into (C-8) yields (assuming 5 < e )
                                          2T  = 0.                    (D-2)
     (D-2) can be transformed into a standard form of Kummer's equation by
(Abramowitz and Stegun, 1964)
                                                                        (D_3)
     Eq. (D-2) becomes
                                       46

-------
The general solution to (D-4) is
                                                                        (D-5)
where A2 and AS are constants and the power series,  M(a,b,z),  is defined by
where                   (a)n = a(a+l)  (a+2)...(a+n-1)  and  (a)Q = 1.
In(D-5)     ,       --                                               (D-7)
While the first part of the solution  can  be  expressed  in terms of elementary
functions as
no such simplification appears  to  be  obtainable  for M(a,b,z).
                                
-------
     In order to match solutions, it is necessary to assume that y and
A (defined in Appendix B) are independent of the assumed form of viscosity
used in each model.  Otherwise the c in D-9 and D-8 will not be the same
coordinate.
     The two solutions D-5 and D-9 have four unknown constants; therefore, it
is necessary to specify four constraints on the problem.
     The boundary condition U(0)  = 0 means

                    °   ^   5=0
which furnishes the first condition.   Requiring a continuous solution and
derivative at the matching point ?  requires that

and
     The fourth condition used is that

            max   f 1 , f2  = 1.0.                                         (D-12)
     With the constraints given in D-10 thru D-12, the normalized variation of
velocity deficit in the vertical  can be computed and the lateral variation can
be computed using (C-7).
                                      48

-------
Figure 3.1. The wind tunnel.
             49

-------
Figure 3.2. Scale model vehicles.
               50

-------
    .50
    .45
.40
    .35
   .30
I
    .20
   .15
   .10
   .05
            T      I      I      I      I      I      I      I      T
               A  1 m FROM TEST SECTION ENTRANCE
               O  2m FROM TEST SECTION ENTRANCE
               a  3 m FROM TEST SECTION ENTRANCE
               •  4m FROM TEST SECTION ENTRANCE
                  .4     .6
                                    U, m/s
                                                              40
                                                                   E>*
             I      I      I      I       I      I       I      I     A|
                                  1.0    ' 1.2    1.4     1.6    1.8   '  2.0
  Rgure 3.3. Empty tunnel, mean velocity profiles for four longitudinal positions on
  centeriine.
                                     51

-------
  •50
  .45
.40i
  .35
  .30
  -25
  .20
  .15
  .10
  .05
           i      i      i      i      r
                  A   ON CENTERL1NE

                  O   -0.205 m FROM CENTERLINE

                  O   +0.205 m FROM CENTERLINE
                                                    i      i       r
                                                                     OOA__
                                                                     a
                                                                     a
            III       I       I       I       I       I
     0      .2      .4      .6      .8      1.0     1.2     1.4     1.6     1.8     2.0
                                      U, m/s
Rgure 3.4. Empty tunnel, mean velocity profiles for three lateral positions at a distance
of 1m from the test section entrance.
                                       52

-------
  .44
  .40
 .36
 .32
 .28
  .24
 .20
 .16
 .12
 .08*
 .04
          i      i      i      i      i      i      i      i      i      r
                         QAO
                  0           A   ON CENTERLINE
                               D   -0.205 m PROM CENTERLINE
                               °   +0.205 m FROM CENTERLINE


                 AO
                        G&   O
                        DA     O
             AID

             a&o

            A  DD
             O
              43      O
            a   A  o
              a A   o
               a
                a   A   °0
I      I      I      i      I      I
                                                 1°    f   °|QD
         .002  .004   .006   .008    .010   .012   .014    .016   .018    .020  .022
Rgure 3.5.  Empty tunnel, turbulence intensity profiles for three lateral positions at a
distance of 1m from the test section entrance.
                                     53

-------
                            Ps
        U.
U-, P,
figure 3.6.  Locations of pressure measurements for determining blockage effects.
                                    54

-------
     0.3
 0?,
O~
0.2
     0.1
      .0
                                                AM/AT'
                                            O   0.0650, FLOOR MOVING
                                            Q   0.0024, FLOOR FIXED
                                            A   0.0051, FLOOR FIXED
                                            •   0.0384, FLOOR FIXED
                                            •   0.0650, FLOOR FIXED
                                            A   0.0041, FLOOR FIXED
             -0.4
                     -0.2
0.2
0.4
                                        y/W
      Figure 3.7.  Spanwise variation on Cpr- for various blockage ratios,
                                        55

-------
              I      1      1      I
        1      1
              1       1
      .3
o
     0:2
                     o
                     *
                      \
                           \
                           A\
     0/f
              I
                     A/
                                            O
                                            D
  I      I
                0.0650, FLOOR
                0.0024, FLOOR
                0.0051, FLOOR
                0.0384, FLOOR
                0.0650, FLOOR
                0.0041, FLOOR
                MOVING
                FIXED
                FIXED
                FIXED
                FIXED
                FIXED


                     I
              10"   2x104
4x104
6x104
8x104
                                        Re
10s
          Figure 3.8. Cpg as function of Reynolds number for various
                                      56

-------
      0.3
     0.2:
"07
      0.1
                            0.02
                                                     U LARGE CAR
                                                     O SMALL CAR
0.04
0.06
                        figure 3.9. Variation of Cp-- with blockage ratio.
                                          57

-------
                                         1/32 SCALE VEHICLE
                                       O 1/8 SCALE VEHICLE
                                      u/U.
Rgure 3.10. Vertical profiles of deficit velocity at x/H = 15, y = 0 for 1/8 and 1/32 scale
vehicles.
                                        58

-------
  .20
                                          •   1/8 SCALE VEHICLE

                                          A   1/32~SCALE VEHICLE
                                       y/W
Rgure 3.11. Lateral profiles of deficit velocity at x/H = 15, z/H = 0.5 for 1/8 and 1/32
scale vehicles.
                                     59

-------
    5.0
    4.5
    4.0
    3.5
    3.0
5,  2-5
    2.0
    1.5
    1.0
               A




               A
                    .025
                                           A  1/32 SCALE VEHICLE


                                           O  1/8 SCALE VEHICLE
                                               B
.050
.075
.100
                                      "(u'2)1'2, m/s
                                                                             a
                                                                             i _
                                                                              o
.125
  Rgure 3.12. Vertical profiles of fluctuating velocity component at x/H = 15, y = 0 for

  1/8 and 1/32 scale vehicles.
                                         60

-------
 2.50
 2.25
 ZOO
 1.75
 1.50
 1.00:
  .75
  .50
                                                 i       i
                                           A  1/8 SCALE VEHICLE

                                           O  EHTHEORY
    0     .025     .050    .075    .100    .125    .150    .175    .200     .225    .250

                                        u/ET
Figure 4.1. Vertical profile of velocity deficit at x/H = 15, y = 0 for 1/8 scale block
vehicle and EH theory.
                                        61

-------
   1.5
5
                  i	L
                                 .1                          .2
                                            u/uT
.3
          Rgure 4.2.  Vertical profiles of velocity deficit for 1/32 scale vehicle, y = 0.
                                        62

-------
0.04
                                            40
                                    X/H
60
80    100
      Figure 4.3.  Decay of maximum velocity deficit with downwind distance.
                               63

-------
    1.7'
 T1.6
 I- 1.5
 d
 :*  1.4
 o
    1.3
    1.1
    1.0
       10
           [x/H]
                                      0.25
20
30
40    50  60
80   100
Rgure 4.4 Rate of growth of the vertical length scale as determined
from velocity profiles.
                            64

-------
Figure 4.5. Smoke release showing slow growth of the wake.
                            65

-------
     .020
     .015
     .009

>|c|'.008

   ~ .007

   «•	
   ^.006
    a.005
i   '-004
     .003
     .002
          10
15
                                   -u'v'/[3u],   z/H = 0.5
                                   —
20
 30


x/hT
40     50    60  70   80 90 100
 Rgure 4.6   Measured Reynolds stress  divided  by velocity shear as a function of
 downwind distance.
                                     66

-------
    3.6
    3.2
    2.8
   2.4
    2.0
    1.6
   1.2
    0.8
   0.4
     -0.02
   O DATA


    .ORIGINAL THEORY [ESKRIDGE AND HUNT, 1979]


    .MODIFIED LATERAL LENGTH SCALE THEORY

     HEIGHT DEPENDENT EDDY VISCOSITY THEORY
.02        .04
.06
                                            U/U«
.08        .10       .12
.14
Rgure 4.7. Theoretical curves and data for vertical profile of velocity deficit for x/H = 30 and
  = 0.
                                     67

-------
                     ORIGINAL THEORY [ESKRIDGE AND HUNT, 1979]
                       MODIFIED LATERAL LENGTH SCALE THEORY
                       HEIGHT DEPENDENT EDDY VISCOSITY THEORY
                       OOATA
    .02  -
      0 t-
                                             1.5
                                            y/W
Rgure 4.8. Theoretical curves and data for lateral profile of velocity deficit for x/H = 30 and
z/H = 0.5.
                                     68

-------
Figure 4.9.  Lateral profile of velocity deficit at three longitudinal positions at constant £".
                                       69

-------
     1.6
    1.4!   _
    1.2
      X/H
 *   ^
 •  : 30, z/H = 0.5
 0  i JO	

—  Height dependent eddy
      viscosity theory
      1   -
s
9
3*   0.8
-N.
i
     0.6  —
     0.4   —
     0.2   -
                                           [y/W]/[x/H]
                                                     0.2S
 Figure 4.10.  Theoretical curve and lateral profiles of velocity deficit at constant £ using
 similarity coordinates.
                                               70

-------
    1.8




    tJB



    1.4




    1.2



o
I    1

^r
i

~   .8




     .6




     .4
                                                    x/H
                                                    30
                                                D   60 ~~~
                                               _____  Height dependent eddy
                                                    viscosity theory
       -0.2    0     .2     .4     .6
.8
                                                        K2    .1.4    1.6    1.8
                                       fu/U.]/rx/H1
Rgure 4.11. Theoretical curve and vertical profiles along centerline of velocity deficit using
similarity coordinates.
                                      71

-------
Rgure 4.12. Lateral variation of the correlation of the fluctuating velocity component, v.
                                 72

-------
 1.0
  .9
  .8
  .7
  .6
  .4
                                      I
                    J	I

     10
20
 30
x/H
40     50   60       80    100
Figure 4.13. Length scale determined from fluctuating velocity component, V.
                                     73

-------
     .6
la
     -4
     .3
     .1
                                                x/H
                                              • 15
                                              • 30
                                              A 45
                                              060
                                    y/u
Rgure 4.14. Correlation of the fluctuating velocity component in similarity coordinates.
                                 74

-------
       .007
      .0003
           10
60  70  80  90 100
Figure 4.15.  Longitudinal variation of the maximum values of the turbulent kinetic energy
components iP and \P together with least squares linear fit
                                 75

-------
                                            H/2
in
CNI
oi
§
         CM
12
d
§
d
m
2
                                                                                                   CO
                                                                                                   0>
                                                                                                   «
                                                                                                   f
                                                                                                   (Q
                                                                                         S
                                                                                         9
                                                                                                   0>
                                                                                                   §
                                                                                                   Q.

                                                                                                   I
                                                                                                   g
                                                                                                   o>
                                                                                         8
                                                                                                   •o
                                                                                                   I
                                                                                                   (0
                                                                                                   o
                                                                                                    o
                                                                                                   3=
                                                      ii
                                                    gz
                                                    2~><
                                                   o::
                                                      (B
                                                   
-------
in
        §
§
8
d
                                                                         8"
                                                                         9
                                                                                  I


                                                                                  1
                                                                                  (B

                                                                                  £
                                                                                  V
                                                                         8   5
                                                                         *r   >•
                                          £
                                          o>

                                          2


                                          «
                                                                                 •o
                                                                         S
                                                                         8
                                                                         oi
                                                                                  O
                                                                                 SB
                                                                                    <0
                                                                                  O> 
-------
H/z
        78

-------
                               I
                               T3

                              !!
                               O
                               O)
                              i-a
79

-------
    3.0
    2,5
    2.0
     1.0
     0.5
                             I       f
               I       I
1I       I        1


               x/H

            A  15

            •  30

            •  45

            O  60

            O  80
I        I       I        I
        0    .0005   .0010    .0015   .0020  .0025   .0030   .0035   .0040    .0045   .0050
Figure 4.20a. Measured centerline vertical of the turbulent kinetic energy component u  at
various downwind distances.
                                  80

-------
    .9
    .8
    .7
1
    -
I5
    .4
    .1
            I



               x/H

           A   15

           «   30


           •   45

           A   60


           0   80
                .02
                    .04
.06
.08
.10
.12
  Rgures 4.20b. Measured centeriine vertical variation of u'2 plotted in TKEC similarity
  variables compared to eq. 4.21 [solid line].
                                81

-------
   3.0
   2,5
   2.0
*  1.5   -
   1.0
   0.5  —
             .0005   .0010   .0015    .0020   .0025    .0030   .0035   .0040   .0045   .0050
                                           2/l|2
                                          V'Z/U
 Rgure 4.21 a.   Measured centeriine vertical variation of the turbulent kinetic energy
 component V2 at various downwind distances.
                                     82

-------
    .8
    .7
     .6
d
     .4
     .1
                                  A  _1_5__

                                  •  30	

                                  •  45     ^

                                  A  60

                                  O  80

                                 	  eq 4.21      _
                   .02
.04
.06
.08
.10
.12
 Figure  4.21 b.  Measured centeriine vertical variation of v'2 plotted in TKEC similarity
 variables compared to eq. 4.21  [solid line].
                                    83

-------
  2.5
        i  i i  I i  i r IT ri i  i I i i  i i I  i i  i i  I I  i 1  i I  i i i  i i  i i
     0     .0005    .001   .0015   .002    .0025    .003    .0035   .004    .0045    .005
Rgure 4.22a.   Measured  centeriine  vertical variation of the turbulent kinetic energy
component w'2 at various downwind distances.
                                 84

-------
     .8
     .7
     .6
  ^
     .4
     .3
     .1
                                 x/H
                               • 15
                               Q 30
                               A 60
                               O 80
                               —i eq 4.21
                   .02
.04
.06
.08
.10
.12
Rgure 4.22b.  Measured centerline vertical variation of w'2 plotted in TKEC similarity
variables compared to eq. 4.21 [solid line].
                                    85

-------
  .0036
                                                     D! 30, z/H = O.S

                                                     O  45

                                                     •  60
                                          y/W
Figure 4.23a. Measured lateral variation of the turbulent kinetic energy component u75 at
various downwind distances.
                                     86

-------
                                                                 x/H
                                                             A   15
                                                             •   30, 2/H = 0.5
                                                             •   45
                                                             D   60
                                                            — eq4.21
                                 I
I
I
I
        0      .15      .30      .45       £0       .75      .90      1.05     1.20    1.35
                                          y/[W[x/H]04]
Figure 4.23b.   Measured  lateral  variation of u'2 plotted in TKEC similarity variables
compared to eq. 4.21 [solid line].
                                        87

-------
   .0025
   .0015   —
I?
                                            y/W
           Figure 4.24a. Measured lateral variation of w  at various downwind distances.

-------
   .12
                                                                     x/H

                                                                 A.  15

                                                                 •  30, Z/H = 0.5

                                                                 •  45

                                                                 A  60

                                                                —  eq4.21
                                                     A  A  A
                .15      .30      .45       .60       .75       .90      1.05     1.20     1.35

__   	  y/[w[x/H]04]	       	
 Figure 4.24b.   Measured lateral variation of w'2 plotted in TKEC similarity  variables
 compared to eq. 4.21  [solid line].
                                       89

-------
  1   -
.  -0.0004   -0.0002
.0002
.0004     .0006     .0008
.001     .0012
                                   u'wvu2
Rgure 4.25a.  Measured centeriine vertical variation  of u'W at various downwind
distances.
                                   90

-------
    3.2
     2.8
     2.4
x
Z
     1.2
      .8
      .4
                  I        I         I
  A  t

  A
i
 A
                                             x/H
                                         •  15
                                         A  30
                                         A  60
                                         O  80

                                                                         a
                                               ..D
                            0*
        -0.06    -0.04     -0.02     0       .02       .04      .06      .08      .1       .12
  Figure 4.25b.  Measured centerline vertical variation of ITw7 in EH similarity coordinates.
                                       91

-------
       80
      70
      60
       50
  5_   40
   ^
  . I.
       30
       20
       10
                                !        I
 •P
A  «
                                            '      '«
                              1       I       I

                                       x/H
                                   •   15_
                                   A   30
                                   A   60^
                                   O   80
                                I        I       I       I
                                      I       I
-   _   -0.015   -0.01   -0.005    0     .005     .01    .015     .02.    .025     .03    .035
                                      u'w'/
    Figure 4.25c. Measured centeriine vertical variation of u'w' in TKEC similarity coordinates.
                                       92

-------
.0032
.0028   —
.0024  —
.0020  —
.0016  —
.0012
.0008  —
.0004
                                                                   50
60
        Figure 4.26a. Lateral variation of u'W at various downwind distances.
                                   93

-------

                                                      1.5
Figure 4.26b.  Measured lateral variation of u'w  in EH similarity coordinates.
                               94

-------
                                                                  , yi,0]
Figure A-1.  Control volume for determining the couple.
                            95

-------
 8

-«.
3
 i
                                      O DATA

                                      — CUBIC SPLINE FIT
      Figure B-1.  Example of cubic spline fit to velocity profile at x/H = 10, y = 0.
                                       96

-------
     0.01
    0.001
z

z
   0.0001
  0.00001
               I    I   I  I I  111
                                                 III  I  I I I I-
        0.1
1.0
10.0

 N
100.0
1000.6
        Figure C-1. Example of velocity spectra at x/H = 15. Spike near 20 Hz is due to a vibration
        from the moving belt
                                            97

-------
    2.4
    2.2
    2.0
    1.8
    1.6
    1.4
    «
    1.0
    0.8
    0.6
    0.4
    0.2
              1       1
                                    I	O-
                                 I—O
                             h-O
^     —
             0.2     0.4     0.6     0.8    1.0     1.2     1.4     1.6
Rgure C-2. Vertical variation of the eddy viscosity determined from
the spectral data.
                            98

-------
  2.50
  2.25
  2.00
  1.75
  1.50
                  o         —
  1.25
  1.00
  0.75
 0.50 <)—
  0.25
    -0.020     -0.015     -0.010    -0.005
.005       .010      .015      .020
                                     u'w'/[[du/dz]UocH]
Figure C-3.  Vertical variation of the eddy viscosity determined from the Reynolds stress
u'w'.
                                        99

-------
                                    TECHNICAL REPORT DATA
                             {Please read Jmtrucriont on the revene before completing]
 1. REPORT NO.
                              2.
                                                            3. RECIPIENT'S ACCESSION>NO.
 4. TITLE AND SUBTITLE
                                                            S. REPORT DATE
 WAKE OF A  BLOCK VEHICLE IN A SHEAR-FREE BOUNDARY  FLOW
 An Experimental and Theoretical Study
              6. PERFORMING ORGANIZATION CODE
 7. AUTHOR(S)
 R.E. Eskridge and R.S. Thompson
              8. PERFORMING ORGANIZATION REPORT NO

              Fluid Modeling  Report  No.  13
 9. PERFORMING ORGANIZATION NAME AND ADDRESS
 Environmental  Sciences Research Laboratory—RTF, NC
 Office of  Research and Development
 U.S. Environmental Protection Agency
 Research Triangle Park, NC  27711
              10. PROGRAM ELEMENT NO.

               ADTA1D/02-1313 (FY-82)
              11. CONTRACT/GRANT NO.
 12. SPONSORING AGENCY NAME AND ADDRESS
 Environmental  Sciences  Research Laboratory—RTF, .NC
 Office of Research  and  Development
 U.S. Environmental  Protection Agency
 Research Triangle Park, NC  27711	
              13. TYPE OF REPORT AND PERIOD COVERED
               In-house 1/1/79 to 10/1/81
              14. SPONSORING AGENCY CODE

                       .-?
 IS. SUPPLEMENTARY NOTES
 16. ABSTRACT
      The wake  of a  moving vehicle was simulated  using a specially-constructed wind
 tunnel with a  moving floor.  A "block-shaped" model  vehicle was fixed 1n position
 over tbe test-section floor while the floor moved  at the freestream air speed to
 produce a uniform,  shear-free, approach flow.  This  simulates an automobile traveling
 along a straight highway under calm atmospheric  conditions.
      Vertical  and lateral profiles of mean and fluctuating velocities and Reynolds
 stresses in the  wake of the vehicle were obtained  using a hot-film anemometer with
 an X-probe.  Profiles were taken at distances of 10  to 80 model heights downwind.
      A momentum  type wake was observed behind the  block-shaped vehicle.  The wake
 does not have  a  simple self-preserving form.  However, 1t 1s possible to collapse
 the velocity deficit with one length and one velocity scale.
      Two new theories for the velocity deficit are compared to the theory of Eskridge
 and Hunt (1979).  A theory which considered a height-dependent eddy viscosity was
 found to fit the data best.
      Length and  velocity scales were found for the longitudinal variation of the
 turbulent kinetic energy.  The lateral variation Is  described by a two-dimensional
 numerical fit  of the crosswind variation of the  data.
 7.
                                KEY WORDS AND DOCUMENT ANALYSIS
                  DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS
e. COSATI Field/Croup
 8. DISTRIBUTION STATEMENT
           RELEASE TO PUBLIC
                                              19. SECURITY CLASS
                                               UNCLASSIFIED
                           21. NO. OF PAGES
                                116
                                              20. SECURITY CLASS (Thispage)

                                               UNCLASSIFIED
                           22. PRICE
CPA Farm 2220.1 O-73)
                                           100

-------