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APPENDIX B
REVIEW OF MODELS
MODEL SELECTION AND DEVELOPMENT
Models that can be used to predict hydroplaning speeds and the depth of sheet flow
over a roadway surface have been published In the literature. These models are identified and
discussed in detail in this appendix. Several types of models are available for predicting the
depth of sheet flow:
One-d~mensional models:
Two~uneDsional models;
Depth of flow over porous pavements
Porous media flow models; and
Other models.
These models were discussed in the main body of this report, and additional information
regarding these models is presented here.
One-Dimensional Dow Models
Russnam and Ross f34) presented a mode! for one~'mensional flow over highway
pavements based on equations developed by Chezy and Manning for open channel flow. Both
B-1
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equations represent turbulent, one-~unensional, steady-state flow. If the slope of the energy
grade fine is assumed to be equal to the slope of the flow path, they represent uniform flow.
The authors also noted that even at velocities and depths when flow would normally be
considered to be I~nar, the impact of rainfall on the fluid surface created conditions that
were turbulent. The Maurung equation was simplified by assuming a wide channel
approximation where the hydraulic radius, R. is equivalent to the depth of flow. This resulted
In the following equation, Be-:
where:
K (L I'm
h=
S n
h =
K -
1
Depth of flow (cm)
Empirically determined constant
Length of the flow path (m)
Rainfall intensity (cm/h)
m,n - Empirically determined exponents
S = Slope of the flow path (m/m)
The values of K, m, and n were determined from data collected on a rolled asphalt
pavement with chipp~ngs and on a brushed concrete pavement. This led to the general
equation for both pavement surfaces, B-2:
h = 0 0~7 (L i)O-47
so.2o
B-2
(Bar)
(B-2)
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Measurements of Mann~g's roughness coefficient, n, were not made, and there was no
statistical evaluation of equation B-2 comparing it to the observed data. The authors also
offered a discussion of the effect of cross-sIope on flow over pavements, suggesting that the
total length of the flow path could be determined by using the vector sum of the slopes to
determine the direction of flow as presented in equation B-3:
if = ~
where:
11 +(mi
m2
1/2
Lf - Length of the flow path
- Width of the pavement lane
I/m ~- Cross slope of the pavement
I/m2 = Longitudmal slope of the pavement
(slope In Me direction of travel)
The report also presented the results of field tests that were used to develop the
~-3)
previous equations. Although the relationships developed by these authors were not used In
this study, their work does represent a significant contribution to the literature, and their data
were used to verify the models developed during this study.
B-3
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Empirical Flow Models
In 1971 and again in 1979, GalIaway et al. (3, 4~46J at the Texas Transportation
Institute developed a set of empirical equations for predicting water depths on road surfaces.
The equations were one-~mensional, inasmuch as they were developed from data collected
from surfaces with slopes in a single principal direction. The equations, based on a regression
analysis, used plane length, rainfall intensity, texture depth, and pavement slope to predict
water depth. Depths were observed at 20 locations on a pavement section. In the 1971 study,
(3J nine pavement surfaces were used, along with six slopes and five rainfall rates. The 1979
study (4J added observations on Portland cement concrete pavements to those in the 1971
study to improve the regression. The coefficients of determination, r2, were 0.68 and 0.83 for
the regression equations reported In the two studies, respectively. The regression equation
wid1 the highest coefficient of determination and the one representing all 1,059 observations on
the surfaces is shown in the following (Ball:
WET = 0.003726 L0519 i0562 MTDo.l2s
so. -
- - MTD
where
(B4)
WET Water film thickness (in) (1 in-25.4 mm)
Plane length (ft)
1
Rainfall intensifier (in/h)
MTD - Mean texture depth (in)
B4
(1 It =305 mm)
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h
Pavement slope (m/m)
Water depth (in) = WET + MID
To date, Gallaway's work represents the single most comprehensive set of water depth
data collected on different types of pavements. The equation, however, is without a
fundamental resistance variable, such as Manning's n, and has been regressed by combining
pavements with different types of surfaces. These models were not used In this study.
Flow Models Based on Kinematic Wave Equation
The kinematic wave approximation was discussed in ache body of this report (equations
4 through 5). To expand, steady-state or equilibrium conditions correspond to the greatest
depths on a flow surface. Under these conditions, depths have increased to the point that
inflow is equal to outflow. For the steady-state case, the term bu/6t is zero. On impervious
surfaces, such as Portland cement concrete, the infiltration rate is zero, and the term f in
equations 4 and 5 is dropped from the right-hand side of the equation. Thus, if infiltration is
zero, the equation representing the conservation of momentum is:
bu ah i u
fix g fix h Ox fx
The kinematic approximation assumes that the velocity terms In equation B-5 are
negligible (i.e., gradually varied flow) and that the gravitational forces are equal to the
frictional forces. Equation B-5 then reduces to the following, B-6:
B-5
03-5,
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S = Sf (B-6)
The kinematic wave approximation has been shown to be valid for conditions when K is
greater than 20, and the Froude number, NE, is greater than O.S, where:
S L
KOX
=
-
hO NF
and ho = normal depth.
Additionally, when NF is less than 0.5,
N2 K > S
.
Rainfall-induced flow on most pavement surfaces falls within the criteria established by
7)
(B-~)
equations B-7 and Bee. Further development results in equations 6 through ~ as discussed in
the body of this report.
Two-D'mensional Flow Models
Flow on highway pavements is a two~,mensional phenomenon. A vertical component
to flow exists that would add a Bird dimension. Yet, since the fluid depths are so small,
B-6
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variations In flow In the indirection can be averaged, and flow may be accurately represented
by a two-dimensional model. Equations B-9 through B-~1 present the equations of state,
continuity, and conservation of momentum for two-~mensional flow.
Conservation of mass:
Gh + 6(uh) + 6(vh) = i - f =
fix BY
where
h Depth of flow
u - Spatially averaged velocities (x - direction)
v = Spatially averaged velocities (y - direction)
i = Rainfall intensity over the domain
f TnfiItration rate
~Incoming rainfall minus Infiltration into subsurface
Conservation of momentum In the Erection leads to:
(B-9)
bu bu bu dh
_ + u- + v , ~+ g is, = g(Sox ~ Set) (B-IO)
u (i -f)
h
Or cost
~ - ~ ~
h
whereas conservation of momentum In the y-direction leads to:
B-7
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By
8t
where
and
TV TV Oh
U- + V- ~ g- = g (Soy
v (i - f) nr cos6',
h
h
+ vt
- Sly)
u, v, h, i, and f are the same as described in B-9
g -
S Ox, S Oy -
So,, Sly,
vt
ex,ey -
(B-1 1)
Acceleration due to gravity (32.17 ft/s2 or 9.806 m/s2)
Slope of the flow path In the x and y directions, respectively
Slope of the energy grade line In the x and y directions, respectively
Terminal rainfall velocity
Angle of rainfall input with respect to the x- and y-axes
The equations are simplified, as were the one~mensional equations, by negating the
force of raindrop impact and, if appropriate, the infiltration term. In most two~mensional
models the remaking terms are retained, and the partial differential terms are approximated
using either a finite difference or finite element scheme. In general, the system of nonlinear,
partial differential equations has no analytical solution and must be solved by numerical
methods.
Zhang and Cundy (33) developed a two~mension~ model for computing water depths
and velocities on a three~imensional surface. The model allows an inclined plane with
B-8
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irregular topography and is based upon a two-step, explicit solution of the finite difference
approximation of the continuity and momentum equations shown earlier in equations B-9
through Bell. The rainfall intensity was considered constant over a finite length but, like
infiltration, could vary over discrete lengths. The model also allowed spatial variations in
plane characteristics, including surface roughness and topography. Tayfur et al. (47) applied
an implicit solution scheme to Zhang and Cundy's model to improve the number of iterations
necessary to reach equilibrium conditions. The model was applied to a planar surface with a
relatively steep slope (eight percent). Due to the nature of nonlinear equations, stability and
convergence remained a problem. Specifically, instability at lower slopes was noted.
Froehlich (4&~) developed a two~unensional, free-surface model to analyze flows
affecting roadway structures, such as culverts, embankments, and bridges. It has the capacity
to analyze unsteady, nonuniform flows but is limited to situations where flow enters or leaves
the flow domain at the boundaries and cannot account for infiltration or flows due to a spatially
distributed source, such as rainfall, on a pavement surface. Huebaer (49) developed a two-
dimensional, steady-state, finite element flow model for flow over highway pavements with
irregular topography. The program produced acceptable results for planar surfaces but
encountered problems with stability and convergence when irregular topographies were
Introduced. The two- dimensional models are cited here for completeness. They were
considered too cumbersome for use In the drain age guidelines as developed for this project.
B-9
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TRIS PAGE ~ENUONALLY LEFT BLAT
Representative terms from entire chapter:
flow models
