GVF Profiles

Open GVF Profiles

Compute water-surface profiles for gradually-varied flow in open channels — backwater curves, drawdown profiles, and flow transitions — across five channel geometries. This guide covers the governing ODE, bed-slope classification, the 12 profile types (M1-M3, S1-S3, C1/C3, H2/H3, A2/A3), the Direct-Step and Standard-Step methods, control sections, and a worked M1 backwater example.

Overview

Gradually varied flow (GVF) occurs when the water surface in an open channel changes gradually over a long distance. Depth is not constant — it is not uniform flow — but the longitudinal changes are gentle enough that streamlines remain approximately parallel and the pressure distribution stays hydrostatic. This distinguishes GVF from rapidly varied flow (hydraulic jumps, sudden expansions, weir crests) where the water surface changes sharply over a short reach and pressure is non-hydrostatic.

GVF analysis is essential for a wide range of hydraulic engineering applications:

• Backwater effects: Computing water-surface profiles upstream of dams, weirs, bridges, and culverts.
• Drawdown profiles: Determining how the water surface drops approaching free overfalls or steeper downstream reaches.
• Flow transitions: Analysing depth changes between mild and steep channel reaches, or around gates and controls.
• Channel design: Sizing channels, bridges, and transitions with adequate freeboard for the computed profile.
• Flood inundation mapping: Generating stage profiles along a river reach for floodplain delineation.

The GVF Profiles tool computes water-surface profiles step-by-step along a channel reach, producing detailed plots and tabular output. All calculations run in your browser — no server round-trip or credits required.

When to use GVF

If the water surface is approximately parallel to the bed over a long reach, use Channel Analysis for uniform-flow conditions. Where the water surface diverges from the bed — approaching a dam, passing over a grade break, emerging from a gate — use GVF profile computation instead.

The GVF Equation

The governing differential equation for gradually varied flow describes how depth changes with distance along the channel. It is derived by combining the energy equation with the continuity equation and differentiating with respect to distance:

$$\frac{dy}{dx} \;=\; \frac{S_0 - S_f}{1 - \mathrm{Fr}^{2}}$$

Governing ODE for gradually varied flow

Where:

• $y$ = flow depth (m)
• $x$ = distance along the channel (m, positive downstream)
• $S_0$ = bed slope — the longitudinal gradient of the channel bottom (m/m)
• $S_f$ = friction slope — the energy gradient, evaluated from Manning’s equation
• $\mathrm{Fr}$ = Froude number = $V / \sqrt{g \cdot D_h}$, where $D_h = A/T$ is the hydraulic depth

The friction slope is computed from the Manning equation rearranged as:

$$S_f \;=\; \left( \frac{n \, Q}{A \, R^{2/3}} \right)^{2}$$

Friction slope from Manning's equation

where $n$ is Manning’s roughness coefficient, $Q$ is the discharge, $A$ is the flow area, and $R$ is the hydraulic radius.

Key relationships

• When $S_f = S_0$, the numerator is zero and $dy/dx = 0$ — this is uniform flow at normal depth.
• When $\mathrm{Fr} = 1$, the denominator is zero — this is critical flow, a singularity in the equation that separates subcritical and supercritical regimes.
• The sign of $dy/dx$ depends on whether the flow is subcritical or supercritical, and whether the actual depth is above or below normal and critical depth. This signed behaviour gives rise to the 12 classical profile types.

Slope Classification

Channel slopes are classified by comparing the normal depth $y_n$ (the depth at which uniform flow would occur) to the critical depth $y_c$ (the depth at which specific energy is minimised for a given discharge). This classification determines which profile types are possible.

Slope Type	Condition	Normal-flow regime	Possible Profiles
Mild (M)	$y_n > y_c$	Subcritical	M1, M2, M3
Steep (S)	$y_n < y_c$	Supercritical	S1, S2, S3
Critical (C)	$y_n = y_c$	Critical	C1, C3
Horizontal (H)	$S_0 = 0$	No uniform flow	H2, H3
Adverse (A)	$S_0 < 0$	No uniform flow	A2, A3

Horizontal and Adverse slopes

For horizontal and adverse slopes, normal depth does not exist because uniform flow is impossible when the bed slope is zero or negative. Only two profile types exist for each, both referenced to critical depth alone.

Profile Types

There are 12 possible GVF profile types, classified by slope type and by the zone in which the flow depth falls. The zones are defined relative to the normal-depth and critical-depth reference lines:

• Zone 1 — above both $y_n$ and $y_c$.
• Zone 2 — between $y_n$ and $y_c$.
• Zone 3 — below both $y_n$ and $y_c$.

Mild Slope Profiles (M1, M2, M3)

On a mild slope, normal depth exceeds critical depth ($y_n > y_c$), meaning uniform flow is subcritical. All three M-profiles are commonly encountered in engineering practice — M1 backwater curves are perhaps the most common GVF computation.

Profile	Depth Range	Physical Scenario	Behaviour
M1	$y > y_n > y_c$	Backwater upstream of a dam, weir, or culvert	Depth decreases in the downstream direction, asymptotic to normal depth upstream
M2	$y_n > y > y_c$	Drawdown approaching a free overfall or steeper reach	Depth decreases in the downstream direction toward critical depth
M3	$y_n > y_c > y$	Supercritical flow downstream of a sluice gate on a mild bed	Depth increases downstream, ending in a hydraulic jump to subcritical flow

Steep Slope Profiles (S1, S2, S3)

On a steep slope, critical depth exceeds normal depth ($y_c > y_n$), meaning uniform flow is supercritical.

Profile	Depth Range	Physical Scenario	Behaviour
S1	$y > y_c > y_n$	Upstream of a dam on a steep slope (following a hydraulic jump)	Depth increases upstream; downstream end controlled by obstruction
S2	$y_c > y > y_n$	Entrance region where flow accelerates from critical toward normal depth	Depth decreases downstream, approaching normal depth asymptotically
S3	$y_c > y_n > y$	Downstream of a sluice gate on a steep slope	Depth increases downstream, approaching normal depth asymptotically

Critical Slope Profiles (C1, C3)

On a critical slope, normal depth equals critical depth ($y_n = y_c$). There is no Zone 2 because the two reference lines coincide. Only C1 and C3 profiles exist.

Profile	Depth Range	Physical Scenario	Behaviour
C1	$y > y_n = y_c$	Upstream of an obstruction on a critical slope	Depth increases upstream; approaches a horizontal asymptote
C3	$y < y_n = y_c$	Downstream of a sluice gate on a critical slope	Depth increases downstream toward critical/normal depth

Critical slope instability

Critical slope conditions are inherently unstable in practice. The flow oscillates between subcritical and supercritical regimes, producing surface undulations and air entrainment. Avoid designing channels at or near the critical slope — always shift the design toward mild or moderately steep.

Horizontal Slope Profiles (H2, H3)

On a horizontal slope ($S_0 = 0$), normal depth is infinite and uniform flow cannot occur. Only two profiles exist, both referenced to critical depth. There is no H1 profile because the water surface cannot lie above an infinite normal depth.

Profile	Depth Range	Physical Scenario	Behaviour
H2	$y > y_c$	Subcritical flow on a flat bed approaching a free overfall	Depth decreases continuously in the downstream direction
H3	$y < y_c$	Supercritical flow on a flat bed (downstream of a sluice gate)	Depth increases downstream, ending in a hydraulic jump

Adverse Slope Profiles (A2, A3)

On an adverse slope ($S_0 < 0$), the channel bed rises in the direction of flow. Like horizontal slopes, normal depth does not exist. Only two profiles exist, both referenced to critical depth.

Profile	Depth Range	Physical Scenario	Behaviour
A2	$y > y_c$	Subcritical flow on an upward-sloping bed	Depth decreases in the downstream direction
A3	$y < y_c$	Supercritical flow on an upward-sloping bed	Depth increases downstream, ending in a hydraulic jump

Adverse slopes in practice

Adverse slopes are rare in natural channels but do occur over short reaches — behind bridge abutments, in channels approaching a higher-elevation stilling basin, or in a localised bed hump. Always check whether the adverse reach is long enough for GVF to develop; very short humps are better treated as rapidly varied flow.

Computation Methods

Because the GVF equation cannot be solved analytically for most channel geometries, numerical step methods are used. The tool supports two complementary approaches, each with distinct advantages.

Direct-Step Method

The Direct-Step method divides the depth range into equal increments and calculates the distance increment for each step. Starting from a known depth at a control section, each depth step produces a distance step using the energy equation:

$$\Delta x \;=\; \frac{E_2 - E_1}{S_0 - \bar{S}_f} \qquad \text{where} \quad E = y + \frac{V^{2}}{2g}$$

Direct-Step distance increment

Here $E$ is the specific energy at each end of the step and $\bar{S}_f$ is the average friction slope over the interval. Key characteristics:

• No iteration required — each step is a direct calculation.
• Produces variable distance spacing (equal depth steps).
• Best suited for prismatic channels (constant cross-section along the reach).
• Fast and numerically stable, but depth discretisation concentrates points near asymptotes.

When to use Direct-Step

Use the Direct-Step method for uniform prismatic channels where you need a smooth depth-distance profile. It is the default and recommended method for most analyses.

Standard-Step Method

The Standard-Step method divides the channel reach into equal distance increments and iteratively solves for the depth at each station. At each station, an assumed depth is refined (by Newton-Raphson or bisection) until the energy equation balances within a specified tolerance:

$$E_1 + z_1 \;=\; E_2 + z_2 + \bar{S}_f \, \Delta x$$

Standard-Step energy balance

where $z$ is the bed elevation. Key characteristics:

• Requires iteration at each step.
• Produces uniform distance spacing (variable depth steps).
• Handles non-prismatic channels where geometry varies with distance — natural river reaches, flared transitions, varying Manning’s $n$.
• More versatile but computationally more expensive.

When to use Standard-Step

Use the Standard-Step method when you need results at specific stations along the channel, or when the channel geometry changes along its length. It is the industry-standard approach implemented in HEC-RAS for natural river reaches.

Channel Types

The GVF Profiles tool supports five channel geometries. Hydraulic properties (area, wetted perimeter, top width) are computed automatically at each depth step.

Geometry	Parameters	Typical Application
Trapezoidal	Bottom width, left & right side slopes	Irrigation canals, drainage channels, roadside ditches
Rectangular	Bottom width	Concrete-lined channels, flumes, box culverts
Triangular	Left & right side slopes	Roadside ditches, small drains, V-notch flumes
Circular	Pipe diameter	Stormwater pipes, culverts, sewers flowing partially full
Custom	Station-elevation pairs	Natural channels, surveyed cross-sections

For detailed geometry formulas see the Channel Analysis guide.

Control Sections

A control section is a location where the flow depth is known or can be determined independently of the GVF calculation. The computation must proceed away from the control in the direction that information propagates. The propagation direction depends on the flow regime.

Flow Regime	Froude Number	Control Location	Computation Direction
Subcritical	$\mathrm{Fr} < 1$	Downstream	Upstream (against flow)
Supercritical	$\mathrm{Fr} > 1$	Upstream	Downstream (with flow)

Computation direction matters

Computing in the wrong direction produces numerical instability and divergent results. The tool automatically determines the correct computation direction based on the specified control and profile type — but the user is responsible for selecting a physically valid control location.

Common control sections include:

• Dam, weir, or spillway crests — critical depth at the crest for an unsubmerged structure.
• Sluice-gate outlets — known depth at the vena contracta from the gate opening and contraction coefficient.
• Channel junctions — depth matched to the downstream tributary or main channel.
• Free overfalls — critical depth at the brink (approximately $0.715 \, y_c$ at the actual brink, but $y_c$ is used at the control).
• Hydraulic jumps — known sequent-depth relationship linking upstream and downstream depths.

Using the Tool

Follow these steps to compute a GVF profile:

1. Select the channel geometry — trapezoidal, rectangular, triangular, circular, or custom station-elevation — and enter the required dimensions.
2. Enter hydraulic parameters — design discharge $Q$, Manning’s $n$, and bed slope $S_0$.
3. Set the boundary condition — specify the known depth at the control section (e.g. the depth at a dam, weir crest, or gate opening).
4. Choose the computation method — Direct-Step or Standard-Step — and set the number of steps.
5. Run the calculation — the tool computes normal and critical depth, classifies the bed slope and profile type, then steps through the reach.
6. Review results — inspect the water-surface profile chart, tabular output per station, and summary hydraulic parameters.

Choosing the number of steps

Start with 50–100 steps for a smooth profile. Increase to 200+ steps if you need higher resolution near a control section or where depth changes rapidly. More steps improve accuracy with minimal computation-time penalty in the browser.

Interpreting Results

The tool produces a water-surface profile chart and a detailed per-station results table. The key outputs:

• Water-surface profile chart — the channel bed, water surface, normal-depth line, and critical-depth line plotted against distance. The shaded region between bed and water surface represents the flow cross-section.
• Normal-depth line ($y_n$) — the dashed reference representing uniform flow depth. In many profile types the water surface asymptotically approaches this line.
• Critical-depth line ($y_c$) — the dashed reference representing critical-flow depth. The water surface approaches this line near free overfalls and regime transitions.
• Froude-number variation — check that the Froude number stays consistent with the expected profile type. A crossing of $\mathrm{Fr} = 1$ indicates a regime transition (often a hydraulic jump).
• Energy grade line — the total energy $E + z$ at each station. A consistently declining energy line confirms the calculation is physically valid.

Watch for profile termination

If the computed profile reaches critical depth ($\mathrm{Fr} = 1$), the GVF equation becomes singular and the numerical integration terminates. This is physically correct — it flags the location where a hydraulic jump, free overfall, or regime transition occurs. The tool reports this condition in the results and identifies it on the chart.

Worked Example — M1 Backwater Upstream of a Weir

The following example demonstrates an M1 backwater calculation using the Direct-Step method.

Given information

• Channel: Trapezoidal, bottom width $B = 4$ m, side slopes $Z = 2H{:}1V$
• Discharge: $Q = 15$ m³/s
• Manning’s n: 0.030 (natural earth channel)
• Bed slope: $S_0 = 0.0010$ (0.10%, mild)
• Control: A broad-crested weir 1 km downstream raises the depth at the weir to 2.50 m
• Aim: Compute the backwater profile and find where depth returns to within 1% of normal depth

Step 1 — Compute normal and critical depth

Iterating the Manning equation gives $y_n \approx 1.60$ m for $Q = 15$ m³/s in this channel. The critical depth from $Q^2 T / (g A^3) = 1$ is $y_c \approx 1.10$ m. Because $y_n > y_c$ the slope is mild.

Step 2 — Classify the profile

The depth at the control (2.50 m) exceeds $y_n$ (1.60 m), so we are in Zone 1 on a mild slope. This is an M1 profile — a classic backwater curve.

Step 3 — Choose computation direction

Flow is subcritical ($\mathrm{Fr} < 1$), so the control is at the downstream weir and computation proceeds upstream. Set the starting depth $y = 2.50$ m at $x = 0$ (at the weir) and integrate toward negative $x$.

Step 4 — Apply the Direct-Step method

Divide the depth range 2.50 m → 1.62 m (1% above normal) into 50 equal increments of $\Delta y = 0.0176$ m. For each sub-reach, compute the average specific energy, average friction slope, and $\Delta x$:

$$\Delta x \;=\; \frac{(y_2 + V_2^{2}/2g) - (y_1 + V_1^{2}/2g)}{S_0 - \tfrac{1}{2}(S_{f,1} + S_{f,2})}$$

Direct-Step increment — numerical form

Step 5 — Summarise the result

Summing the 50 sub-reach distances gives a total backwater length of approximately 1 650 m. The water surface is elevated above normal depth throughout this reach — with about 0.90 m extra depth at the weir, reducing smoothly to about 0.02 m at the upstream end.

Practical takeaway

Backwater from a fairly modest 0.9 m rise at the weir extends over 1.6 km upstream on this mild slope. Backwater is highly sensitive to bed slope — halving $S_0$ roughly doubles the backwater length. Always compute the full profile before sizing upstream freeboard, bridges, or riverine infrastructure.

Hydraulic Jumps

Where a supercritical profile meets a subcritical control downstream, the flow cannot pass through critical depth gradually — it transitions abruptly via a hydraulic jump. The GVF equation does not describe the jump itself (that is rapidly varied flow), but the two sides of the jump are connected by the momentum equation:

$$\frac{y_2}{y_1} \;=\; \tfrac{1}{2}\left(\sqrt{1 + 8\,\mathrm{Fr}_1^{2}} - 1\right)$$

Sequent depth relationship (rectangular channel)

where $y_1$, $\mathrm{Fr}_1$ are the supercritical upstream depth and Froude number and $y_2$ is the conjugate subcritical depth. In practice, the jump location is found by computing the upstream supercritical profile, the downstream subcritical profile, and locating the station where the supercritical depth’s conjugate equals the subcritical water-surface depth.

Jumps and energy loss

A hydraulic jump dissipates a significant fraction of the kinetic energy — up to 85% for $\mathrm{Fr}_1 > 9$. This makes the jump a useful energy dissipator (stilling basins), but it also means the energy grade line drops sharply across the jump. Size downstream linings to survive the turbulent region immediately after the jump.

References

• Chow, V.T. (1959). Open-Channel Hydraulics. McGraw-Hill, New York. The foundational reference for GVF profile classification and computation — Chapters 9 and 10 are the seminal treatment.
• Henderson, F.M. (1966). Open Channel Flow. Macmillan, New York. Comprehensive treatment of gradually varied flow theory with extensive worked examples.
• Chaudhry, M.H. (2008). Open-Channel Flow (2nd ed.). Springer, New York. Modern presentation of numerical methods including the Standard-Step method and Newton-Raphson iteration.
• French, R.H. (1985). Open-Channel Hydraulics. McGraw-Hill, New York. Practical guide to open-channel flow analysis with emphasis on engineering applications.
• USACE. (2016). HEC-RAS River Analysis System Hydraulic Reference Manual (Version 5.0). US Army Corps of Engineers Hydrologic Engineering Center, Davis, CA. Reference implementation of the Standard-Step method in industry practice.
• SANRAL. (2013). Drainage Manual (6th ed.). South African National Roads Agency, Pretoria. Chapter 7 — Hydraulic design of open channels and backwater analysis.

---

Open GVF Profiles