What is a Head-discharge

Head-discharge refers to the hydraulic relationship between the rate of water flow (discharge) and the driving water level (head) that causes or influences that flow. It is a fundamental principle in open channel hydraulics and is used to model and design structures such as weirs, orifices, culverts, outfalls, and flow control devices in drainage and water management systems.

The term is typically expressed as a mathematical or empirical relationship that correlates the upstream water level (head) with the corresponding flow rate (discharge), often written as:

Q = f(H)

Where:

  • Q is the discharge (typically in litres per second or cubic metres per second)
  • H is the head (height of water above a reference point, often in metres)
  • f(H) represents the functional form of the relationship, which depends on the flow structure and flow conditions

Understanding the head-discharge relationship is essential for accurately sizing and predicting the performance of hydraulic structures under varying flow scenarios.

Key Concepts

1. Head

Head refers to the height of the water column above a specific point or structure that exerts pressure or energy driving the flow. It can include:

  • Static head: Vertical distance between water surface and outlet
  • Velocity head: Associated with the kinetic energy of moving water
  • Total head: The combination of elevation, pressure, and velocity head

In head-discharge calculations, upstream water depth relative to the structure is usually the key variable.

2. Discharge

Discharge is the volume of water flowing past a point per unit time. It is typically measured in:

  • Litres per second (L/s)
  • Cubic metres per second (m³/s)

The discharge depends not only on the head but also on the geometry and type of structure, flow conditions, and coefficients of discharge.

Typical Applications of Head-Discharge Relationships

1. Weirs

For a sharp-crested weir, the head-discharge relationship often takes the form:

Q = C × L × H^1.5

Where:

  • C is the discharge coefficient (depends on weir type and flow conditions)
  • L is the length of the weir
  • H is the head above the weir crest

This equation allows engineers to predict how much water will pass over a weir for a given upstream depth.

2. Orifices

For orifice plates or culverts under submerged or free-flow conditions:

Q = C × A × √(2gH)

Where:

  • A is the cross-sectional area of the opening
  • g is the acceleration due to gravity
  • C is the orifice discharge coefficient
  • H is the head above the orifice centreline

3. Outfall Controls and Hydrobrakes

Flow control devices such as vortex flow regulators are designed with specific head-discharge curves to restrict or modulate flow depending on water levels, often to comply with SuDS or flood control standards.

Importance in Design and Modelling

Accurate head-discharge relationships are critical for:

  • Hydraulic structure sizing
  • Flood risk assessments
  • Drainage system modelling
  • Sustainable Drainage Systems (SuDS) performance verification
  • Flow regulation and attenuation system design

Most design software (e.g. MicroDrainage, InfoWorks ICM, SWMM) includes libraries of standard head-discharge relationships, or allows engineers to input custom curves based on laboratory testing or manufacturer data.

Factors Affecting Head-Discharge Relationships

Several factors can influence the accuracy and applicability of head-discharge formulas:

  • Flow regime (free-flow vs. submerged flow)
  • Inlet and outlet conditions
  • Approach velocity
  • Structure geometry (shape, size, roughness)
  • Discharge coefficient (depends on empirical calibration)

Site-specific calibration or physical modelling may be required for complex or non-standard structures.

Measurement and Verification

In operational systems, head-discharge relationships can be measured or verified using:

  • Level sensors (ultrasonic, pressure transducers)
  • Flow meters (e.g. electromagnetic or weir flow measurements)
  • Calibration charts or tables based on actual performance data

Accurate field data helps refine design assumptions and improve system performance evaluations.

Conclusion

The head-discharge relationship is a cornerstone of hydraulic engineering, describing how water levels influence flow rates through various drainage and conveyance structures. Understanding this relationship enables engineers to design effective flood control, attenuation, and sewerage systems that perform reliably under variable loading conditions. Whether in the form of a simple formula or a complex curve derived from modelling, head-discharge data underpins sound decision-making in water infrastructure design and operation.