
(4.19)
A transverse Eulerian integral length scale defined by
^vjjjv^ j (420)
o V'(o)
was computed from data obtained using two xfilm 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 xdependence 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 powerlaw 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 vehiclegenerated 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 3min. averaging period. The function F (x,o>) was determined by a
w
leastsquares 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 ±00 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 crosssections 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 selfpreserving wakes starting with order one, where mean
velocity profiles are selfpreserving; order two, where in addition Reynolds
stresses are also selfpreserving 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 selfpreserving 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 TWODIMENSIONAL 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. 1518, Reno, NV. Am. Meteorolooical Soc., Boston, MA,
p 58491.
Bevilaqua, P.M. and P.S. Lykoudis, 1978: Turbulence memory in self preserving
wakes. J_. Fluid Mech., 89, 589606.
Castro, I.P., 1979: Relaxing wakes behind surfacemounted obstacles in rough
wall boundary layers. J_. Fluid Mech.. 9_3, 631659.
Castro, I.P. and J.E. Fackrell, 1978: A note on twodimensional fence flows,
with emphasis on wall constraint. J_. Indust. Aero., _3, 120.
Castro, I.P. and A.G. Robins, 1977: The flow around a surfacemounted cube
in uniform and turbulent streams. J_. Fluid Mech., 79, 307335.
Clark, R.E., R.N. Kubik, and L.P. Phillips, 1963: Orthogonal polynomial least
squares surface fit. Comm. ACM., 6_, 146148.
Counihan, J., J.C.R. Hunt and P.S. Jackson, 1974: Wakes behind twodimensional
surface obstacles in turbulent boundary layers. J_. Fluid Mech., 64, 526563.
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'Iii 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., 387400.
Goldstein, S., 1938: Modern Developments in Fluid Dynamics. Oxford University
Press, Oxford. Reprinted by Dover.
Hansen, A.C. and J.E. Cermak, 1975: Vortexcontaining wakes of surface obstacles,
Project THEMIS Technical report no. 29. (Fluid Dynamics and Diff. Lab. Rpt.No.
CER7576ACHJFC16,) Colorado State University. Ft. Collins, CO. 163 pp., Dec.
Hinze, J.O., 1975: Turbulence. McGrawHill, 79C pp.
Hunt, J.C.R., 1971: A theory for the laminar wake of a twodimensional body
in a boundary layer. J. Fluid Mech., 49, 159178.
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.. !]_, 291297.
Schlichting, H., 1960: Boundary Layer Theory, Sixth edition, McGrawHill,
New York, NY., 747 pp.
Smith, F.T., R.I. Sykes and P.W.M. Brighton, 1977: A twodimensional boun
dary layer encountering a threedimensional hump. J_. Fluid Mech., 83, 163176.
Snyder, W.H., 1979: The EPA Meteorological Wind Tunnel  Its Design, Constru
ction, and Operating Characteristics. EPA 600/479051, 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 CR2806, 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. Al . 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 (Uu,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 yaxis, the dot product of j" with
(Al) is taken; this, together with an assumption of constant density gives
(A2)
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 yaxis 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 yaxis 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 (A2) 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(xi, y, z) = 0. The T'S may also be considered perturbation quantities
since TUDstream = ^ (uniform flow). Equation A3 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 (A2).
Face (1)
a 
also / x? v;? \. "f  zU and
therefore I =  f 2 U
' ^j j
o X
30

Face (2)
also trx£) = 2 UutvV,,*)  XW(X,V;,E);
and
Xl
I 2 = 
Face (3)  avv(
v(i f s
also (rx)«J = 2 iyu)Xj.w ^J r\.? = Uu . Hewce,
I3 = f
o ^
and
also
and (w«= V
Thus T4r f
*< o
31