---
title: "Explain GVF and spatially varied flow"
subject: "Hydraulics"
chapter: 5
key: "semester_4/hydraulics:gvf-and-svf-explain-example"
source: https://www.bctengineer.com/q/semester_4/hydraulics/gvf-and-svf-explain-example
site: BCT Engineering
---

# Explain GVF and spatially varied flow

Hydraulics · Chapter 5: Basics of Open channel flow

- Asked in: 2069 Bhadra, 4 marks
- Video: [Gradually varied flow & Rapidly varied flow | Types of Non Uniform Flow | GVF | RVF l Open Channel](https://www.youtube.com/watch?v=YaNsgt9r60E) — Professor Nidhi Trivedi
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/gvf-and-svf-explain-example
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-5

## Question

Explain gradually varied flow and spatially varied flow with one practical example for each.

## Short answer

Gradually varied flow (GVF) is steady non-uniform flow with constant discharge and slowly changing depth; e.g., backwater curve behind a dam. Spatially varied flow (SVF) has discharge varying along the channel due to lateral inflow or outflow; e.g., a side channel spillway (increasing Q) or a side weir (decreasing Q).

## Answer

#### Gradually varied flow (GVF)
GVF is steady non-uniform flow in which the depth changes slowly from section to section over a long distance. The streamlines are nearly parallel, so the pressure is hydrostatic and the friction loss can be found by applying a uniform-flow formula (Manning or Chezy) locally. The discharge is constant: $dQ/dx=0$.

Governing (dynamic) equation:
$$
\frac{dy}{dx}=\frac{S_0-S_f}{1-Fr^2}
$$
**Example:** the backwater curve upstream of a dam or weir (the depth rises gradually towards the weir); the drawdown curve in a mild channel approaching a free overfall.

```text
    backwater (GVF)
  ~~~~~~~___-----~~~~|
  y_n               ||  weir
  ____________________||
```

#### Spatially varied flow (SVF)
SVF is steady flow in which the discharge varies along the channel length because water is added or removed along the way, so $dQ/dx\neq0$. For increasing discharge (lateral inflow), the momentum equation is used, because the mixing of the incoming water causes an unknown energy loss. For decreasing discharge (outflow over a side weir), the energy equation is usually used, because the outflow causes little energy loss.

Dynamic equation for lateral inflow (with negligible momentum of the inflow in the flow direction): $\dfrac{dy}{dx}=\dfrac{S_0-S_f-\dfrac{2Q}{gA^2}\dfrac{dQ}{dx}}{1-Fr^2}$.

**Example:** a side channel spillway or a roadside gutter (increasing Q); a side weir or a canal with several outlets (decreasing Q).

| Feature | GVF | SVF |
|---|---|---|
| Discharge | constant | varies with $x$ |
| Depth | changes slowly | changes slowly or fast |
| Cause | change in slope, section or control | lateral inflow/outflow |

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which statement describes gradually varied flow?
   A. Steady flow with depth changing slowly over a long distance and constant discharge
   B. Flow with depth changing abruptly over a short distance
   C. Flow with discharge changing along the length
   D. Flow with constant depth everywhere
   Answer: A. GVF is steady non-uniform flow where depth changes slowly, streamlines are nearly parallel and Q is constant. (Key point: GVF definition)

2. The GVF dynamic equation is dy/dx = (S0 - Sf)/(1 - Fr^2). What does the numerator show?
   A. Depth changes only when Fr = 1
   B. Depth is unchanged only when the bed slope equals the friction slope
   C. Depth always rises when Sf is positive
   D. The friction slope has no effect
   Answer: B. If S0 = Sf the numerator is zero and dy/dx = 0, which is uniform flow; otherwise depth changes. (Key point: GVF equation)

3. What is the key difference of spatially varied flow from GVF?
   A. Depth changes more slowly
   B. The channel is not prismatic
   C. Discharge varies along the length (dQ/dx is not zero) due to lateral inflow or outflow
   D. The flow is unsteady
   Answer: C. In SVF water is added or removed along the reach, so Q changes with x. (Key point: SVF definition)

4. Why is the momentum equation usually used for SVF, and which is an example with decreasing Q?
   A. Lateral flow causes unknown energy loss at the junction; side weir
   B. Friction is zero; side channel spillway
   C. Depth is constant; roadside gutter
   D. Energy is conserved exactly; backwater curve
   Answer: A. Lateral flow causes an unknown energy loss, so momentum is used. A side weir removes flow, so Q decreases. (Key point: SVF method and examples)

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