Scope and the quantity being calculated #

Superelevation is the transverse difference in free-surface elevation produced when flowing water follows a horizontal curve. The outside water surface rises because of centrifugal forces that pinch the water against the outer wall. The formula for superelevation water level can be found in handbooks such as FHWA, USACE, and USBR. However, they might have different formulas and coefficients, so which one is right?

In this article we will trace through the physics of superelevation of water level around curve conduits and try to derive the formula from first principles. We also show how to use it as a geometric screening calculation for a line-based channel network connected through a two-dimensional sump. In that application, the sump footprint supplies the length scale missing from the inlet and outlet lines, and the resulting radius is a declared geometric reference radius rather than a measured hydraulic streamline radius.

However, we do not directly calculate a water level in a completely full, pressurised pipe. A full pipe has no internal free surface. The same radial momentum balance then predicts a transverse pressure or piezometric-head difference, not an inner-to-outer water-surface difference. Nonetheless, as we are working with gravity pipe networks, this is not an issue concerning us.

First, two definitions:

\displaystyle \Delta h_{io}=\eta_o-\eta_i

is the full difference from the inner wall to the outer wall, whereas

\displaystyle \Delta h_o=\eta_o-\eta_c

is the rise at the outer wall above a specified centerline or reference surface. For a narrow, symmetric channel under equilibrium conditions,

\displaystyle \Delta h_o\approx \frac{1}{2}\Delta h_{io}.

It’s easy to overlook the distinction when you are in the heat of the moment or just plain tired, therefore it’s important to keep this distinction in mind.

The diagram uses the same convention as the equations: \Delta h_{io}=\eta_o-\eta_i is the complete transverse difference, while \Delta h_o=\eta_o-\eta_c is the rise used by the USACE centerline-to-outside formulation.

Derivation from the governing physics #

Coordinates and assumptions #

Let:

• s be distance along the curved channel centerline;
• n be transverse distance measured outward from that centerline;
• z be vertical elevation;
• R be the horizontal radius of the channel centerline;
• B be the water-surface width;
• u(n) be the streamwise velocity at transverse position n ; and
• \eta(n) be the free-surface elevation at transverse position n — the quantity we’re solving for.

At position n , the local streamline radius is approximately

\displaystyle r=R+n.

For the basic derivation, we assume steady incompressible flow, a single-valued free surface, negligible vertical acceleration, approximately concentric streamlines, and negligible transverse velocity and shear in the leading-order transverse balance.

Hydrostatic pressure #

Negligible vertical acceleration gives

\displaystyle \frac{\partial p}{\partial z}=-\rho g.

With atmospheric pressure at the free surface z=\eta(n) ,

\displaystyle p(n,z)=p_{atm}+\rho g[\eta(n)-z].

Therefore, at a fixed elevation,

\displaystyle \frac{\partial p}{\partial n}=\rho g\frac{d\eta}{dn}.

Transverse momentum balance #

Water moving at speed u around a streamline of radius r requires inward acceleration u^2/r — where does that come from? Only the pressure field can supply it, so pressure must increase toward the outside of the bend. The leading-order transverse momentum equation is

\displaystyle \frac{1}{\rho}\frac{\partial p}{\partial n}=\frac{u(n)^2}{R+n}.

Combining this with the hydrostatic relation gives the fundamental free-surface equation:

\displaystyle \boxed{\frac{d\eta}{dn}=\frac{u(n)^2}{g(R+n)}}.

This differential equation is the physical result, and every compact engineering formula that follows is just this equation with added assumptions about velocity and geometry bolted on.

So the pressure gradient points outward while the acceleration of the moving water points inward — opposite vectors, equal in the leading-order radial balance.

Transverse integration #

Across a channel extending from n=-B/2 to n=+B/2 ,

\displaystyle \boxed{ \Delta h_{io} =\int_{-B/2}^{B/2}\frac{u(n)^2}{g(R+n)}\,dn }.

This form shows why real velocity distribution matters: superelevation depends on u^2 , not merely on discharge or mean velocity.

If the streamwise velocity is approximated as uniform, u(n)=V=Q/A , the integral becomes

\displaystyle \boxed{ \Delta h_{io} =\frac{V^2}{g} \ln\!\left(\frac{R+B/2}{R-B/2}\right) }.

This logarithmic result retains the geometric variation in streamline radius but still assumes a uniform velocity and equilibrium flow.

Velocity-distribution examples #

We include the classical forced- and free-vortex cases here because they are useful limiting examples: they show exactly where the assumed transverse velocity distribution enters the water-surface shape. Neither should be presumed to describe an ordinary channel bend without supporting evidence.

For a forced vortex, or solid-body rotation,

\displaystyle u(r)=\omega r,

so

\displaystyle \eta(r)=\frac{\omega^2r^2}{2g}+C_0, \qquad \Delta h_{io}=\frac{\omega^2}{2g}\left(r_o^2-r_i^2\right).

The free surface is parabolic in r . For a free vortex,

\displaystyle u(r)r=K_v,

so

\displaystyle \eta(r)=C_0-\frac{K_v^2}{2gr^2}, \qquad \Delta h_{io}=\frac{K_v^2}{2g} \left(\frac{1}{r_i^2}-\frac{1}{r_o^2}\right).

The free surface is inverse-square in radius, and the uniform-velocity assumption we use for the engineering formula sits algebraically between these two idealised descriptions:

\displaystyle u(r)=V \quad\Longrightarrow\quad \eta(r)=C_0+\frac{V^2}{g}\ln r.

Narrow-channel approximation #

For B/R\ll1 ,

\displaystyle \ln\!\left(\frac{R+B/2}{R-B/2}\right) =\frac{B}{R}+\frac{1}{12}\left(\frac{B}{R}\right)^3+O\!\left[\left(\frac{B}{R}\right)^5\right].

Keeping only the leading term gives

\displaystyle \boxed{\Delta h_{io}\approx\frac{V^2B}{gR}}.

For a symmetric narrow section, the outside-wall rise relative to the theoretical centerline surface is approximately

\displaystyle \boxed{\Delta h_o\approx\frac{V^2B}{2gR}}.

Equivalently, for a rectangular section with hydraulic depth D=A/T=y (A the cross-sectional flow area and T the top width — the same quantity as B here, since the section is rectangular),

\displaystyle \frac{\Delta h_{io}}{y} \approx Fr^2\frac{B}{R}, \qquad Fr=\frac{V}{\sqrt{gD}}.

This dimensionless form makes the controlling physics visible: transverse tilt grows with the square of Froude number and with curvature B/R .

How handbook forms relate to the derivation #

Why do the FHWA and USACE formulas look different? Mostly bookkeeping — they define the reported rise differently, not the physics behind it.

FHWA: full inner-to-outer difference #

FHWA HEC-22 gives, for subcritical open-channel flow around a bend,

\displaystyle \boxed{\Delta d=\frac{V^2T}{gR_c}},

where R_c is the radius to the channel centerline, T is water-surface width, and \Delta d is the full difference between inner and outer banks. It states that the outer surface is approximately \Delta d/2 above the centerline surface and the inner surface is approximately \Delta d/2 below it. This is the narrow-channel result above. See FHWA HEC-22, Fourth Edition, Section 6.1.5.

USACE: outside-wall rise including specified wave allowance #

USACE EM 1110-2-1601 defines

\displaystyle \boxed{\Delta h_o=C\frac{V^2W}{gR}},

as the rise between a theoretical level centerline surface and the outside water surface, where W is the channel width at the centerline water surface. Its coefficient C depends on flow regime, cross-section type, and curve treatment:

Flow condition	Curve treatment	Rectangular C	Trapezoidal C
Tranquil	Simple circular	0.5	0.5
Rapid	Simple circular	1.0	1.0
Rapid	Spiral transitions	0.5	1.0
Rapid	Spiral and banked invert	0.5	Not tabulated

So C=0.5 for a tranquil simple curve is consistent with taking half of the inner-to-outer equilibrium difference, and the rapid-flow coefficients simply add an allowance for the greater surface rise and standing-wave behaviour seen in those configurations — they are not universal turbulence or pipe-shape factors, so don’t reach for them outside the flow conditions they were tabulated for. See USACE EM 1110-2-1601, Section 2-5 and Table 2-4.

Urban-drainage software policy for C #

For the preliminary sump screen developed here, we fix

\displaystyle \boxed{C_{\mathrm{ref}}=0.5}

for every estimate that it returns, provided that the calculation is restricted to a free-surface, subcritical state. This makes \Delta h_{\mathrm{ref}} a centerline-to-outside equilibrium-rise reference consistent with the tranquil-flow USACE coefficient; it also avoids presenting C as a user-adjustable calibration parameter that the available one-dimensional model cannot determine.

The policy must be enforced as

\displaystyle C_{\mathrm{ref}}= \begin{cases} 0.5, & \text{free surface, }Fr\lt 1,\ R_{\mathrm{ref,geom}}\gt B_{\mathrm{ref}}/2,\\ \text{not evaluated}, & \text{otherwise}. \end{cases}

Urban drainage doesn’t guarantee tranquil flow on its own. Part-full steep conduits can be supercritical, and a surcharged sump can lose its free surface, so the software must withhold this reference estimate for Fr\ge1 , pressurised flow, or unavailable radius geometry — it must not keep using 0.5 and imply that rapid-flow waves or pressure effects have been covered.

For a two-dimensional sump, C_{\mathrm{ref}}=0.5 remains an explicit reference convention extrapolated from a smooth tranquil curve; it is not an experimental calibration of the sump. Consequently the output remains a Preliminary Reference Water-Level Rise, not a predicted maximum.

In a fully pressurised pipe, there is no internal free surface, so \Delta h_{\mathrm{ref}} cannot represent water-level superelevation and C_{\mathrm{ref}} is not assigned. Curvature instead produces a transverse pressure or piezometric-head gradient, which must be evaluated as a pressurised-flow problem.

Conditions required for validity #

The compact equation holds up best when all of the following are approximately true:

1. A free surface exists. A completely pressurised pipe requires a pressure-field analysis instead.
2. The flow is steady or changes slowly enough to be treated as quasi-steady. Rapid hydrograph changes add local acceleration.
3. Vertical pressure is approximately hydrostatic. Splashing, plunging jets, wall run-up, abrupt vertical curvature, and strong vertical acceleration violate this assumption.
4. The bend is smooth and its curvature varies gradually. An elbow, miter bend, chamber, or abrupt 90-degree junction is a momentum-turning problem, not a continuous-curvature equilibrium bend.
5. Streamlines approximately follow the bend. Strong separation or a large recirculation zone invalidates the assumed radius.
6. The velocity distribution is known or acceptably represented by V . Boundary layers and secondary circulation alter the u^2 -weighted integral.
7. The section has had enough bend length to approach transverse equilibrium. At bend entry and exit, waves and transient transverse motion may govern instead.
8. For direct use of the simple formula, the flow is subcritical and not close to a strongly unstable or wave-dominated state. In supercritical flow, disturbances propagate as oblique and reflected waves, so the maximum wall level is not generally the equilibrium tilt.
9. The required result is a mean or design water-surface rise, not splash height or an instantaneous local maximum. Those require empirical wave allowances, physical modelling, or higher-fidelity numerical modelling.

Applicability to urban drainage design #

For urban drainage, a conduit may be part-full with a free surface early in the event and later become surcharged; flow through a sump may also reverse or switch between competing inlet-outlet paths. So we have to evaluate applicability at each reporting timestep — we can’t just assign it once from the conduit type or plan angle and forget about it.

For the urban-drainage screening method in this document, a numerical reference rise may be returned only when a continuous free surface exists, Fr\lt 1 , one dominant turning path can be identified, B_{\mathrm{ref}} is defined from the active free-surface width or declared geometric proxy, and the constructed radius satisfies R_{\mathrm{ref,geom}}\gt B_{\mathrm{ref}}/2 . The fixed value C_{\mathrm{ref}}=0.5 then represents the tranquil-flow centerline-to-outside convention; it does not calibrate the local sump hydraulics.

A typical urban sump, access chamber, or manhole is an abrupt chamber rather than a smooth curved channel. Even when the qualification above is satisfied ( free surface and all), the calculated \Delta h_{\mathrm{ref}} can still only be used as a preliminary reference water-level rise. It is not an exact outside-wall level, an overtopping certificate, or a substitute for a project-specific freeboard requirement.

The condition B/R\ll1 is the asymptotic assumption used to replace the exact logarithmic uniform-velocity result by the compact V^2B/(gR) expression. Because “much less than” has no unique numerical cutoff, it is not by itself a binary software gate. The cited recommendation R/B\ge3 (equivalently B/R\le1/3 ) is used as the practical smooth-bend screen. For 0.5\lt R_{\mathrm{ref,geom}}/B_{\mathrm{ref}}\lt 3 in subcritical flow, the software may preserve the calculated reference value but must issue a strong warning. If the ratio is at least 3, the geometry is only not rejected; passing the ratio does not demonstrate attached flow, hydrostatic pressure, transverse equilibrium, or absence of waves and separation.

The calculation must be withheld ( “no result”), rather than reported as zero, when R_{\mathrm{ref,geom}}/B_{\mathrm{ref}}\le0.5 , because the assumed inner radius R-B/2 is then nonpositive and the logarithmic curved-section geometry is undefined. It must also be withheld when the sump is fully pressurised, Fr\ge1 , the active flow path is ambiguous, a plunging jet or drop dominates, flow reversal invalidates the selected path, or no finite radius or width can be defined. Where the available rim margin or consequence depends on the actual local level, the appropriate next step is full-momentum 2D modelling, measurements or physical modelling, or validated 3D CFD when vertical jets, air entrainment, impact, or wall pressure are important.

What limits apply to geometric R/B ? #

Here R should be the plan radius to the conveyance centerline and B the water-surface width at the evaluated flow — geometric quantities once the operating water depth fixes the top width, and neither one is the hydraulic radius A/P , whatever a quick skim of the formula might suggest.

Handbook quantities and sump proxies #

The handbook quantities should not be described as “hydraulic radius and hydraulic width.” FHWA HEC-22, Section 6.1.5 defines R_c as the radius to the channel centerline and T as the channel surface width. USACE EM 1110-2-1601, Section 2-5 and USBR Design Standards No. 14, Chapter 3 similarly define r as the radius of channel-centerline curvature and W or T as the channel width at the elevation of the centerline water surface. Therefore, the handbook radius is an alignment-geometry quantity, while the width is the free-surface top width at the evaluated hydraulic state. Neither radius is the hydraulic radius A/P .

Quantity	Direct handbook use in a continuous curved channel	Urban-sump screening substitute
Radius	Actual radius of channel-centerline curvature	R_{\mathrm{ref,geom}} from the largest tangent circular arc accommodated by the inlet direction, outlet direction, and usable sump footprint
Width	Actual free-surface width at the evaluated water level	B_{\mathrm{ref}} , preferably calculated as the active top width from the current depth and section geometry; if only a clear geometric width is available, it must be labelled as a proxy

For a smooth curved channel, we use the handbook quantities directly. But an abrupt sump has no constructed curved channel centerline and may not even have a single attached turning stream. The handbooks do not state that a largest tangent arc fitted inside a sump is equivalent to their channel-centerline radius, and they do not validate the superelevation equation for this substitution, neither does our simplified physics equation tell us how to proceed in this case. However, we still use the 2D sump profile on plan as a useful construct to derive the R .

The convention is useful because it is reproducible from the available two-dimensional geometry and avoids inventing a radius from the zero-radius line intersection. Selecting the largest accommodated tangent arc gives the highest R_{\mathrm{ref,geom}}/B_{\mathrm{ref}} and the smallest \Delta h_{\mathrm{ref}} among the admitted circular reference paths. Consequently, failing the ratio screen is informative: even the most favourable geometric construction is too sharp. However, passing the screen is weaker; it only means the chosen proxy is not rejected and does not establish that the water follows that arc.

We do this not only because this is the only (cheap) clue we have, but also because the whole thing is an approximation anyway. Calculating it more precisely will not increase the overall precision by much.

Thus, software output must report the source of each value—handbook channel geometry, depth-derived top width, or sump-derived geometric proxy—and retain the label Preliminary Reference Water-Level Rise whenever a sump proxy is used.

Mathematical limit #

The logarithmic uniform-velocity integral requires

\displaystyle R\gt \frac{B}{2},

so that the inner streamline radius remains positive. This is only a coordinate and geometry requirement, not evidence that the engineering formula is reliable down to R/B=0.5 .

Let \lambda=R/B . After cancelling the common factor V^2/g , the exact logarithmic and compact geometric factors are

\displaystyle L(\lambda)=\ln\!\left(\frac{\lambda+1/2}{\lambda-1/2}\right), \qquad A(\lambda)=\frac{1}{\lambda}.

The percentage by which the compact result underestimates the logarithmic result is calculated with the logarithmic result in the denominator:

\displaystyle \varepsilon_{\mathrm{low}}(\lambda)=100\,\frac{L(\lambda)-A(\lambda)}{L(\lambda)}=100\left[1-\frac{1/\lambda}{\ln\!\left((\lambda+1/2)/(\lambda-1/2)\right)}\right]\%.

For example, at R/B=2 , L=\ln(2.5/1.5)=0.510826 , A=0.5 , and \varepsilon_{\mathrm{low}}=2.119\% . The complete comparison is:

R/B	Compact result relative to the logarithmic uniform-velocity result
10	0.08% low
5	0.33% low
3	0.93% low
2	2.12% low
1	8.98% low
0.5	Singular

The small approximation error at R/B=2 does not prove hydraulic validity there, since velocity redistribution, secondary circulation, separation, and bend-entry waves can all be much larger than the Taylor-series error.

Published design guidance #

There is no universal handbook range stating that the formula is valid for every R/B . Existing guidance is conditional:

• USACE, tranquil/subcritical flow: R/B\ge3 is suggested for design to minimise helicoidal flow and velocity distortion. This is the clearest general geometric recommendation in EM 1110-2-1601.

• USBR, subcritical conveyance: current spillway and outlet-work guidance repeats a centerline curvature of at least three times the channel or structure width. See USBR Design Standards No. 14, Chapter 3.

• FHWA HEC-22: the subcritical bend equation is given without an explicit R/B validity limit. That absence must not be interpreted as approval for arbitrarily sharp bends.

• USACE, rapid/supercritical rectangular flow: a constant R/B limit is not used. For curves with spiral transitions, USACE reports the minimum centerline radius

  \displaystyle R_{min}=\frac{4V^2B}{gy}.

  For a rectangular section, this becomes

  \displaystyle \frac{R_{min}}{B}=4Fr^2.

  Supercritical design also calls for transition geometry and wave treatment; it is not justified by satisfying this radius equation alone.

• USDA SCS TR-25: for subcritical trapezoidal channels it uses a section-dependent expression rather than a universal R/B coefficient,

  \displaystyle s=\frac{V^2(b+2zd)}{2(gR-2zV^2)},

  where s is the superelevation rise (TR-25’s own symbol — not the along-channel distance s used earlier in this document), b is the channel bottom width, z is the side-slope ratio, horizontal to vertical (again TR-25’s own notation, distinct from the vertical-elevation z defined earlier), and d is the flow depth. It recommends limiting the rise to the lesser of 1 ft or 10% of water-surface width. For supercritical lined channels it provides separate transition guidance and warns against trapezoidal sections on curved alignments because wave run-up is difficult to predict. See USDA SCS Technical Release 25.

A workable practical interpretation is therefore:

Geometry and flow	Treatment of the simple equation
Subcritical, smooth curve, R/B\ge3	Appropriate preliminary or handbook calculation when the other assumptions hold
Subcritical, smooth curve, 1\lt R/B\lt 3	Sensitivity or screening estimate only unless supported by applicable guidance, measurements, or modelling
0.5\lt R/B\le1 , abrupt elbow, junction, or chamber	Do not treat the equilibrium formula as a design prediction; a sump-derived reference radius may still support a clearly labelled screening calculation with a strong warning
R/B\le0.5	Withhold the curved-section result because the assumed inner radius is nonpositive
Supercritical flow	Apply the relevant rapid-flow wave and transition method; do not use the tranquil-flow equation alone
Fully pressurised pipe	Calculate pressure/piezometric-head distribution and bend forces, not free-surface superelevation

These categories are an engineering interpretation of the formula and cited guidance, not a new code limit.

Deriving a reference radius from a two-dimensional sump #

Why the network vertex is not the whole geometry #

If we treat an inlet line and an outlet line as nothing more than mathematical line segments meeting at one non-collinear vertex, their local centerline radius is zero, and substituting into V^2B/(gR) gives an undefined result — proof enough that a smooth-bend equilibrium equation cannot be applied at the abstract vertex itself.

A physical sump contains more geometry. The inlet channel terminates at one connection on the sump boundary, the flow occupies a finite two-dimensional chamber, and the outlet begins at another connection. The inlet and outlet axes may intersect virtually, but water is not constrained to turn at that zero-dimensional intersection. The sump footprint therefore provides the missing length scale from which a finite, reproducible reference turn can be constructed.

This construction does not discover the actual streamline radius. It answers a narrower geometric question:

What is the largest circular centerline turn, tangent to the inlet and outlet directions, that the available sump plan geometry can accommodate?

We denote that quantity by R_{\mathrm{ref,geom}} . Put plainly: it is the friendliest possible bend the sump’s footprint could physically fit, not the bend the water is actually taking.

Required plan quantities #

Project the three-dimensional sump and connected channels onto the horizontal plane and obtain:

• the usable sump footprint S ;
• the inlet connection midpoint P_i ;
• the outlet connection midpoint P_o ;
• the inlet centerline direction \mathbf t_i , directed into the sump; and
• the outlet centerline direction \mathbf t_o , directed out of the sump.

The flow-deflection angle is

\displaystyle \theta=\cos^{-1}\!\left(\operatorname{clamp}(\mathbf t_i\cdot\mathbf t_o,-1,1)\right),

where \theta=0 is straight-through flow. For nonparallel axes, extend the inlet and outlet centerlines until they meet at the virtual point of intersection PI . The available distances along the two axes are

\displaystyle T_i=|P_i-PI|, \qquad T_o=|P_o-PI|.

These distances exist because the sump connections occur at finite positions on a two-dimensional footprint, and we cannot get them from channel width and bend angle alone.

We construct the center C and the red tangent arc from the plan geometry, not by smoothing the abstract line intersection at PI .

Symmetric sump #

For a circular arc tangent to two lines separated by deflection angle \theta , the tangent length is

\displaystyle T=R\tan\left(\frac{\theta}{2}\right).

If the two available distances are equal, T_i=T_o=T , a single circular reference arc can be tangent to both axes at the connection locations:

\displaystyle \boxed{ R_{\mathrm{ref,geom}} =\frac{T}{\tan(\theta/2)} }.

For a 90-degree turn, \tan45^\circ=1 , so

\displaystyle \boxed{R_{\mathrm{ref,geom}}=T}.

For example, if the axes of two perpendicular channels intersect at the center of a 400 mm square sump and each connection midpoint is 200 mm from that intersection, the reference radius is 200 mm.

Asymmetric sump #

If T_i\ne T_o , one circular arc cannot generally be tangent to both axes at both fixed connection midpoints. A fixed, repeatable single-radius convention is to use the shorter available tangent distance:

\displaystyle \boxed{ R_{\mathrm{candidate}} =\frac{\min(T_i,T_o)}{\tan(\theta/2)} }.

The resulting arc uses all the available distance on the shorter side and leaves a straight extension on the longer side. It must still be checked against the complete sump footprint. If any part of the candidate arc falls outside S , reduce the radius until the whole centerline arc lies within S .

For an irregular sump, the general definition is therefore

\displaystyle \boxed{ R_{\mathrm{ref,geom}} =\max\left\{R:\mathcal A(R,\theta)\subseteq S\right\}, }

where \mathcal A(R,\theta) is a circular arc tangent to the inlet and outlet axes, with tangent points restricted to the available portions of those axes inside the sump. The optimization is geometric; it does not require a CFD velocity field.

If the sump contains internal obstructions, benching, columns, or excluded regions, they must be removed from S before testing arc containment. If a finite tangent arc cannot be constructed, report the radius as unavailable and withhold the reference \Delta h ; do not substitute zero or a small numerical tolerance into the denominator.

Selecting the reference width #

The radius construction and the width definition must be reported separately. For direct use in a smooth open channel, B is the water-surface width at the evaluated flow. For the sump screen, the software may use a declared geometric proxy such as the connected-channel clear width or the active water-surface width calculated from depth:

\displaystyle B_{\mathrm{ref}}=B_{\mathrm{channel,geom}} \quad\text{or}\quad B_{\mathrm{ref}}=T(y).

For a rectangular channel these are normally the same. For a trapezoidal section or partially full circular conduit, water-surface width varies with depth and should be calculated if the hydraulic state is available. Using chamber width and connected-channel width as two scenarios produces two reproducible evaluations, not lower and upper bounds on the real hydraulic result.

The geometric curvature ratio is

\displaystyle \boxed{ \lambda_{\mathrm{ref}} =\frac{R_{\mathrm{ref,geom}}}{B_{\mathrm{ref}}} }.

Interpretation of the warning #

Because R_{\mathrm{ref,geom}} is defined as the largest accommodated tangent-arc radius, it is the most favorable circular-turn geometry available under the stated construction:

• If \lambda_{\mathrm{ref}}\le0.5 , withhold the result because the assumed inner radius is nonpositive.
• If \lambda_{\mathrm{ref}}\ge3 , the geometry is only not rejected by that screen. It does not prove that flow in the sump is attached, hydrostatic, concentric, or accurately represented by the equilibrium formula.
• For applicable rectangular rapid-flow screening, compare \lambda_{\mathrm{ref}} with 4Fr^2 , while retaining the separate requirement for rapid-flow transition and wave assessment.

The corresponding reference rise is

\displaystyle \boxed{ \Delta h_{\mathrm{ref}} =C_{\mathrm{ref}}\frac{V_{\mathrm{ref}}^2B_{\mathrm{ref}}} {gR_{\mathrm{ref,geom}}} =0.5\frac{V_{\mathrm{ref}}^2B_{\mathrm{ref}}} {gR_{\mathrm{ref,geom}}} }.

Since the construction deliberately selects the largest admissible radius and \Delta h\propto1/R , this is the smallest equilibrium-rise estimate among the circular reference arcs admitted by the construction. It is not a conservative maximum and must not be reported as the actual outside-wall level.

Suitable result names are Sump-Derived Geometric Radius, Geometric Curvature-Ratio Screen, and Preliminary Reference Water-Level Rise. Suitable warning text is:

The radius is derived from the largest tangent circular arc accommodated by the two-dimensional sump footprint. It is a geometric reference, not a resolved streamline radius. Failure of the handbook R/B criterion flags geometry outside the ideal smooth-bend range; passing it does not validate the sump-flow assumptions.

Implementation sequence #

1. Confirm that a free surface exists at the evaluated state.
2. Project the sump footprint and channel connection geometry into plan.
3. Determine P_i , P_o , \mathbf t_i , \mathbf t_o , PI , and \theta .
4. Calculate the symmetric candidate radius from the available tangent distances.
5. Reduce the candidate if necessary until the entire tangent arc is contained in the usable sump footprint.
6. Declare and calculate B_{\mathrm{ref}} .
7. Calculate R_{\mathrm{ref,geom}}/B_{\mathrm{ref}} and compare it with the applicable flow-regime criterion.
8. Calculate \Delta h_{\mathrm{ref}} only when R_{\mathrm{ref,geom}}\gt B_{\mathrm{ref}}/2 , retaining its preliminary-reference label.
9. Escalate to full-momentum 2D or validated 3D modelling when the decision depends on actual local wall level, splash, pressure, or structural load.

Evaluation through a dynamic event #

We must evaluate the reference rise from one physically consistent hydraulic state at a time — never combine the maximum HGL from one timestep with the maximum velocity from another. For every reporting time that passes the free-surface and subcritical gates, we calculate

\displaystyle \Delta h_{\mathrm{ref}}(t) =0.5\frac{V_{\mathrm{ref}}(t)^2B_{\mathrm{ref}}(t)} {gR_{\mathrm{ref,geom}}},

and, when a containment comparison is required,

\displaystyle Z_{\mathrm{outside,ref}}(t) =H_{\mathrm{node}}(t)+\Delta h_{\mathrm{ref}}(t).

The governing timestep is

\displaystyle t_*=\operatorname*{arg\,max}_{t\in\mathcal T_{\mathrm{valid}}} Z_{\mathrm{outside,ref}}(t),

where \mathcal T_{\mathrm{valid}} contains only timesteps with a free surface, Fr(t)\lt 1 , a defined flow path and width, and R_{\mathrm{ref,geom}}\gt B_{\mathrm{ref}}(t)/2 . Peak discharge, peak velocity, peak node HGL, and the maximum reference outside level need not occur simultaneously — a steady calculation evaluates only the state the engineer selected, while a dynamic calculation can search all valid reporting states without mixing them.

How a two-dimensional solver calculates superelevation #

Governing equations #

A depth-averaged two-dimensional hydraulic solver normally solves the shallow-water equations over the plan geometry:

\displaystyle \frac{\partial h}{\partial t}+\nabla\cdot(h\mathbf{u})=0,

\displaystyle \frac{\partial(h\mathbf{u})}{\partial t} +\nabla\cdot\left(h\mathbf{u}\otimes\mathbf{u}+\frac{1}{2}gh^2\mathbf{I}\right) =-gh\nabla z_b+\mathbf{S}_{f}+\mathbf{S}_{t},

where h is depth, \mathbf{u} is depth-averaged horizontal velocity, z_b is bed elevation, \mathbf{S}_f represents bed friction, and \mathbf{S}_t represents modelled turbulent or dispersive stresses.

In Cartesian coordinates the solver does not need to be given a separate centrifugal-force formula — the channel walls and mesh turn the velocity vector for it. The advective momentum-flux term

\displaystyle \nabla\cdot(h\mathbf{u}\otimes\mathbf{u})

then produces the acceleration associated with that turn, while the gravity-pressure term produces a transverse water-surface gradient. At a resolved steady bend, projecting the numerical momentum balance normal to the streamline recovers approximately

\displaystyle g\frac{\partial\eta}{\partial n}\approx\frac{u_s^2}{R},

with additional friction, turbulent-stress, unsteady, and nonuniform-flow contributions retained by the model.

Cell-by-cell numerical update #

A finite-volume solver divides the sump and channels into plan cells. For cell i , it stores cell-averaged water depth and horizontal momentum,

\displaystyle \mathbf U_i=(h,hu,hv)_i.

During one time step, it calculates mass and momentum fluxes through every cell face and updates the cell:

\displaystyle \mathbf U_i^{\,n+1}=\mathbf U_i^{\,n}-\frac{\Delta t}{A_i}\sum_f \mathbf F_{if}L_f+\Delta t\,\mathbf S_i.

Here A_i is cell area, L_f is face length, \mathbf F_{if} is the numerical flux through face f , and \mathbf S_i contains bed slope, friction, and other source terms. The flux leaving one cell enters its neighbour, so mass and momentum are transferred conservatively from cell to cell. At a solid wall the normal flux is constrained; around a bend the successive face directions turn the momentum vector. The resulting imbalance between advective momentum and pressure flux changes h across the width, producing different cell water-surface elevations \eta_i=z_{b,i}+h_i . Repeat this update through time and the outside-high, inside-low surface develops on its own — nobody imposes a superelevation formula on it.

Finite volume, finite difference, and finite element are alternative spatial discretisations; time stepping advances any of them from n to n+1 . HEC-RAS 2D specifically uses an unstructured finite-volume scheme, not a finite-element scheme.

The colour gradient represents the computed transverse water-surface variation — an output of the full momentum balance, not an imposed V^2B/(gR) correction bolted on afterward.

Minimum numerical requirements #

For a bend-superelevation calculation, the minimum setup that holds up is:

1. Use the full shallow-water momentum equations. Diffusion wave omits convective acceleration and cannot reliably generate bend superelevation. HEC-RAS identifies bend superelevation as a full-momentum application.
2. Resolve the transverse variation. Place approximately 5 to 7 or more active cells across the channel or turning-flow width where detailed velocities and water-surface differences are required, align faces or breaklines with walls and banks, and repeat with a finer mesh. The reported outside-minus-inside level should change by less than the project tolerance.
3. Demonstrate time-step convergence. Reduce \Delta t and confirm that the peak and time history of the outside-wall level are stable; a run that merely completes without instability is not sufficient.
4. Use physical boundaries and a fixed reporting definition. Apply the actual inflow hydrograph or velocity distribution and downstream level, then compare water levels at specified inner and outer locations rather than using an unrestricted single-cell maximum.

See the HEC-RAS guidance on equation selection, computational meshes, and hydraulic property tables and transverse cell resolution.

What two-dimensional modelling still misses #

Depth-averaged shallow-water equations assume hydrostatic pressure and do not resolve the vertical velocity structure. They can represent the plan-view momentum turn, transverse surface tilt, propagation and reflection of depth-averaged waves, and spatially varying depth and velocity. They cannot directly resolve:

• vertical wall run-up and splash;
• air entrainment;
• plunging or impinging jets;
• the vertical structure of secondary helical circulation;
• nonhydrostatic pressure near abrupt obstacles; or
• a transition between a free surface and a completely full pipe unless a specialised mixed-flow treatment is used.

When those effects govern, we need a validated three-dimensional free-surface model — commonly a volume-of-fluid method — or a physical model instead. And a visually convincing CFD surface is not validation on its own: mesh, time-step, turbulence, boundary-condition, and scale sensitivity still have to be demonstrated against conservation checks and, where the consequences justify it, real measurements.

Relevance to present-day hydraulic engineering #

The formula remains useful because it is a transparent momentum balance, not because it replaces modern simulation. It still serves four current purposes:

1. Preliminary design. It provides a rapid estimate of wall rise and a first check on curve radius and freeboard.
2. Screening. It identifies bends where Fr^2B/R is large enough to justify 2D modelling, physical testing, or geometric redesign.
3. Independent verification. A resolved 2D or 3D result should have the correct order of magnitude and trend relative to V^2B/(gR) when the ideal assumptions are approached. A large unexplained disagreement is a reason to inspect the mesh, boundaries, velocity field, or applicability of the formula.
4. Engineering communication. It shows directly how velocity, width, and radius affect risk: doubling velocity quadruples the equilibrium rise, doubling width doubles it, and doubling radius halves it.

For routine subcritical bends with a generous radius, the equation may be enough on its own, combined with the governing handbook and normal freeboard requirements. But for sharp curves, supercritical flow, abrupt junctions, highly three-dimensional structures, or decisions sensitive to the instantaneous maximum wall level, its role changes — from design prediction to benchmark and screening calculation.

Compact decision guide #

For an urban-drainage sump, use the equation as a screening calculation:

Check	Screening response
No free surface, Fr\ge1 , or no identifiable turning path	Do not calculate water-level superelevation; report not applicable and use an appropriate pressure, wave, or higher-dimensional method.
R_{\mathrm{ref,geom}}/B_{\mathrm{ref}}\le0.5	Withhold the result because the assumed curved-section geometry is undefined.
0.5\lt R_{\mathrm{ref,geom}}/B_{\mathrm{ref}}\lt 3	Calculate the preliminary reference rise with C_{\mathrm{ref}}=0.5 , but issue an R/B applicability warning.
R_{\mathrm{ref,geom}}/B_{\mathrm{ref}}\ge3	Calculate the preliminary reference rise with C_{\mathrm{ref}}=0.5 ; no geometric-ratio warning is required.

For every valid timestep, compare H_{\mathrm{node}}+\Delta h_{\mathrm{ref}} with the sump rim or required containment level, and if the reference outside level approaches or exceeds the available level — or the consequence of getting it wrong is significant — escalate to full-momentum 2D modelling, measurements or physical modelling, or validated 3D CFD as appropriate. Passing the screen does not prove the actual local wall level, and it does not certify the design.

Principal references #

• Federal Highway Administration, Urban Drainage Design Manual, HEC-22, Fourth Edition, FHWA-HIF-24-006, Section 6.1.5.
• U.S. Army Corps of Engineers, Hydraulic Design of Flood Control Channels, EM 1110-2-1601, Section 2-5.
• U.S. Bureau of Reclamation, Design Standards No. 14: Appurtenant Structures for Dams, Chapter 3—General Spillway Design Considerations.
• U.S. Department of Agriculture, Soil Conservation Service, Design of Open Channels, Technical Release No. 25.
• U.S. Army Corps of Engineers, HEC-RAS 2D User’s Manual, guidance on shallow-water and diffusion-wave equation sets.