1. Siphonic roof drainage systems with Gutters
Marc Buitenhuis
Hydraulic research engineer Akatherm International BV, Panningen, The Netherlands
02-03-2009
Abstract
On the European continent it is common to install siphonic roof drainage systems directly on
flat roofs. In other parts of the world, like the UK and Australia for example, the rain water is
collected in gutters first.
In this article the theory behind the flow in gutters is described.
1. Introduction: Flow in gutters
q
q
On the European continent it is common to
install siphonic roof drainage systems
directly on flat roofs. In other parts of the
world, like the UK and Australia for v
example, the rain water is collected in D
v + dv
gutters first. β
Usually these gutters are installed under a A
A + dA
small inclination so that the water will flow W
towards the roof outlet, driven by gravity.
The siphonic drainage system will add a
suction force to this through its working Mass continuity
principle.
The flow will also experience counter The mass flowing out of this control
forces: besides the friction force there is an volume must equal the incoming mass
opposing hydrostatic pressure resulting flow, which consists of the mass flow
from the additional amount of water upstream of the control volume plus the
supplied by the runoff from the roof along source term of the runoff from the roof:
the gutter length. 
In this article the equations that describe Q = A( x ) ⋅ v( x ) + q = A( x + ∂x ) ⋅ v ( x + ∂x ) =
gutter flow will be presented. [ A( x ) + ∂A( x )] ⋅ [v( x ) + ∂v( x )] =>
q = A( x ) ⋅ ∂v ( x ) + ∂A( x ) ⋅ v( x ) + ∂A( x ) ⋅ ∂v ( x )
2. Theoretical background
A control volume can be defined for which Momentum equation
the mass and momentum continuity
equations can be derived. Similarly the momentum forces acting on
the control volume can be constructed.

2. The inertia forces of the flow in the length Falling water term
direction of the gutter are constructed
similarly as the mass continuity: The water dropping from the roof in to the

M = −ρ ( x ) ⋅ v( x ) + ρ ( x +∂ ) ⋅
A A
2
x gutter will add its inertia to the control
v ( x +∂ ) 2 = 2 ρ ( x ) ⋅ v ( x ) ⋅ ∂ ( x ) +
x A v
volume system. In the length direction this
accounts for:
ρ A( x ) ⋅ v( x ) 2 +O ( ∂ ) 2
∂ x
Frunoff = ρqv f ⋅ ∂x ⋅ sin (θ ) with
Frictional Force v f = 2 g ( h0 + x sin (θ ) )
The frictional forces acting on the flow are Hydrostatic pressure term
present on the wall boundaries only.
 ∂p  As mentioned above there will be a net
τ
F friction = − w  p + ∂x
 2  static pressure force acting on the control
The wall shear stress can be expressed by: volume caused by the additional amount of
ρv 2 water supplied by the runoff.
τw = f with For an arbitrary cross sectional area with
2
1 the form function L(y) describing the
7.4n 2 width of the gutter at depth y the static
f = 1 / 3 ; n = 0.0382ε 6
RH pressure term can be described by:
Fp = ρg ∫ L( y ) ⋅ y ⋅ dy
D
and RH/ 3 = A / p
1
0
or rewritten:
Cρv 2
τ w = 1 / 3 with C = 3.7n 2
RH
q
q
v
v + dv
β
The net hydrostatic pressure force in the
τw
g length direction is then described by:
dFp d  D
 L( y ) ydy ∂x
dx  ∫0
∂x = ρg 
dx 
Gravitational Force
or:
The gravitational force has a component in dFp d  D dD
the length direction when the gutter is ∂x = ρg  ∫0 L( y ) ydy 
 ∂x
installed under inclination: dx dD   dx
 ∂A 
Fgravitation = ρg  A + ∂x sin (θ ) ≈ Total Momentum equation
 2 
 ∂A  Taking all first order terms and dividing by
ρg  A + ∂x ⋅ S0
 2  ρ∂x the following equation results:
when θ is small and S0 is the slope.

3. (v ∂A + 2 Av∂v ) = ∂( Av ) =
2 2
The maximum depth will be found when
∂x ∂x the nominator is 0:
2
Cv 2 Cp gA3 S0 v f A S0 2 A
− p + gAS 0 + qv f S 0 − − 2 2 − + = 0 =>
RH/ 3
1
RH/ 3
1
q x qx 2 x
d  D dD Cp 2 A  gA3 v f A 
2
 L( y ) ydy 
dD  ∫0
g  + = 2 2 +  S0
 dx RH/ 3
1
x qx 2 
q x
 
q⋅x From this equation the optimum slope of
with v = the gutter can be determined.
A
(v ∂A + 2 Av∂v
2
=
) (
∂ Av 2
=
) The above theory neglects one effect and
∂x ∂x that is the effect of the suction of the
dA ∂D siphonic system. Thus the above equation
v2 + 2qv
dD ∂x is the worst case depth profile.
Substitution in the equation above and 3. Conclusions
∂D
deriving from it gives:
∂x In this article the theory of the flow in
2
Cp gA3 S0 v f A S0 2 A gutters has been described. The theory
− 2 2 − + neglects the effect of suction by the
∂D R1 / 3 q x qx 2 x
= H
siphonic system and thus the equations
∂x dA A g d2
L( y ) ydy
D
dD q 2 x 2 dD ∫0
− predict a worst case depth profile.
4. References
which defines the depth profile along the
gutter. 1. Gwilym T. Still, Flows in Gutters &
When the denominator becomes 0 the Downpipes, 2001, University of Warwick,
school of engineering
equation goes to infinity. In practice the
flow will then go from supercritical to
subcritical and form a hydraulic jump (
v
Fr = = 1 with h=hydraulic
gh
depth=A/surface width).
Also when x tends to zero (far end of
gutter) the solution tends to infinity. So it
is necessary to take an approximation for
the depth at this point.
When the friction or hydraulic pressure
terms increase the gradient of the depth
profile increases. When the slope or falling
water velocity increases the gradient
decreases, i.e. the water is flowing away
easier. From this the conclusions can be
drawn that decreasing the friction or
increasing the slope helps to minimize the
waterlevel in the gutter. The slope will be
limited practically.

4. (v ∂A + 2 Av∂v ) = ∂( Av ) =
2 2
The maximum depth will be found when
∂x ∂x the nominator is 0:
2
Cv 2 Cp gA3 S0 v f A S0 2 A
− p + gAS 0 + qv f S 0 − − 2 2 − + = 0 =>
RH/ 3
1
RH/ 3
1
q x qx 2 x
d  D dD Cp 2 A  gA3 v f A 
2
 L( y ) ydy 
dD  ∫0
g  + = 2 2 +  S0
 dx RH/ 3
1
x qx 2 
q x
 
q⋅x From this equation the optimum slope of
with v = the gutter can be determined.
A
(v ∂A + 2 Av∂v
2
=
) (
∂ Av 2
=
) The above theory neglects one effect and
∂x ∂x that is the effect of the suction of the
dA ∂D siphonic system. Thus the above equation
v2 + 2qv
dD ∂x is the worst case depth profile.
Substitution in the equation above and 3. Conclusions
∂D
deriving from it gives:
∂x In this article the theory of the flow in
2
Cp gA3 S0 v f A S0 2 A gutters has been described. The theory
− 2 2 − + neglects the effect of suction by the
∂D R1 / 3 q x qx 2 x
= H
siphonic system and thus the equations
∂x dA A g d2
L( y ) ydy
D
dD q 2 x 2 dD ∫0
− predict a worst case depth profile.
4. References
which defines the depth profile along the
gutter. 1. Gwilym T. Still, Flows in Gutters &
When the denominator becomes 0 the Downpipes, 2001, University of Warwick,
school of engineering
equation goes to infinity. In practice the
flow will then go from supercritical to
subcritical and form a hydraulic jump (
v
Fr = = 1 with h=hydraulic
gh
depth=A/surface width).
Also when x tends to zero (far end of
gutter) the solution tends to infinity. So it
is necessary to take an approximation for
the depth at this point.
When the friction or hydraulic pressure
terms increase the gradient of the depth
profile increases. When the slope or falling
water velocity increases the gradient
decreases, i.e. the water is flowing away
easier. From this the conclusions can be
drawn that decreasing the friction or
increasing the slope helps to minimize the
waterlevel in the gutter. The slope will be
limited practically.

5. (v ∂A + 2 Av∂v ) = ∂( Av ) =
2 2
The maximum depth will be found when
∂x ∂x the nominator is 0:
2
Cv 2 Cp gA3 S0 v f A S0 2 A
− p + gAS 0 + qv f S 0 − − 2 2 − + = 0 =>
RH/ 3
1
RH/ 3
1
q x qx 2 x
d  D dD Cp 2 A  gA3 v f A 
2
 L( y ) ydy 
dD  ∫0
g  + = 2 2 +  S0
 dx RH/ 3
1
x qx 2 
q x
 
q⋅x From this equation the optimum slope of
with v = the gutter can be determined.
A
(v ∂A + 2 Av∂v
2
=
) (
∂ Av 2
=
) The above theory neglects one effect and
∂x ∂x that is the effect of the suction of the
dA ∂D siphonic system. Thus the above equation
v2 + 2qv
dD ∂x is the worst case depth profile.
Substitution in the equation above and 3. Conclusions
∂D
deriving from it gives:
∂x In this article the theory of the flow in
2
Cp gA3 S0 v f A S0 2 A gutters has been described. The theory
− 2 2 − + neglects the effect of suction by the
∂D R1 / 3 q x qx 2 x
= H
siphonic system and thus the equations
∂x dA A g d2
L( y ) ydy
D
dD q 2 x 2 dD ∫0
− predict a worst case depth profile.
4. References
which defines the depth profile along the
gutter. 1. Gwilym T. Still, Flows in Gutters &
When the denominator becomes 0 the Downpipes, 2001, University of Warwick,
school of engineering
equation goes to infinity. In practice the
flow will then go from supercritical to
subcritical and form a hydraulic jump (
v
Fr = = 1 with h=hydraulic
gh
depth=A/surface width).
Also when x tends to zero (far end of
gutter) the solution tends to infinity. So it
is necessary to take an approximation for
the depth at this point.
When the friction or hydraulic pressure
terms increase the gradient of the depth
profile increases. When the slope or falling
water velocity increases the gradient
decreases, i.e. the water is flowing away
easier. From this the conclusions can be
drawn that decreasing the friction or
increasing the slope helps to minimize the
waterlevel in the gutter. The slope will be
limited practically.

6. (v ∂A + 2 Av∂v ) = ∂( Av ) =
2 2
The maximum depth will be found when
∂x ∂x the nominator is 0:
2
Cv 2 Cp gA3 S0 v f A S0 2 A
− p + gAS 0 + qv f S 0 − − 2 2 − + = 0 =>
RH/ 3
1
RH/ 3
1
q x qx 2 x
d  D dD Cp 2 A  gA3 v f A 
2
 L( y ) ydy 
dD  ∫0
g  + = 2 2 +  S0
 dx RH/ 3
1
x qx 2 
q x
 
q⋅x From this equation the optimum slope of
with v = the gutter can be determined.
A
(v ∂A + 2 Av∂v
2
=
) (
∂ Av 2
=
) The above theory neglects one effect and
∂x ∂x that is the effect of the suction of the
dA ∂D siphonic system. Thus the above equation
v2 + 2qv
dD ∂x is the worst case depth profile.
Substitution in the equation above and 3. Conclusions
∂D
deriving from it gives:
∂x In this article the theory of the flow in
2
Cp gA3 S0 v f A S0 2 A gutters has been described. The theory
− 2 2 − + neglects the effect of suction by the
∂D R1 / 3 q x qx 2 x
= H
siphonic system and thus the equations
∂x dA A g d2
L( y ) ydy
D
dD q 2 x 2 dD ∫0
− predict a worst case depth profile.
4. References
which defines the depth profile along the
gutter. 1. Gwilym T. Still, Flows in Gutters &
When the denominator becomes 0 the Downpipes, 2001, University of Warwick,
school of engineering
equation goes to infinity. In practice the
flow will then go from supercritical to
subcritical and form a hydraulic jump (
v
Fr = = 1 with h=hydraulic
gh
depth=A/surface width).
Also when x tends to zero (far end of
gutter) the solution tends to infinity. So it
is necessary to take an approximation for
the depth at this point.
When the friction or hydraulic pressure
terms increase the gradient of the depth
profile increases. When the slope or falling
water velocity increases the gradient
decreases, i.e. the water is flowing away
easier. From this the conclusions can be
drawn that decreasing the friction or
increasing the slope helps to minimize the
waterlevel in the gutter. The slope will be
limited practically.