Question

A stream channel narrows between support columns under a bridge. If discharge does not change, predict how stream velocity would be altered as water flowed under the bridge. a) Velocity would increase. b) Velocity would decrease. c) Velocity would not change.

Short answer

a) Velocity would increase.

Step-by-step solution

Understanding Fluid Dynamics

When dealing with stream flow, the concept of discharge (Q) is crucial. Discharge is the volume of water flowing past a point in a specific amount of time and is calculated as the product of velocity (V) and cross-sectional area (A) of the stream: \( Q = V \times A \).

Analyzing the Scenario

The problem states that the discharge does not change as the stream narrows under the bridge. However, the reduction in the cross-sectional area of the stream (A) as it gets narrower will affect the velocity.

Applying the Continuity Equation

According to the principle of continuity, if the discharge \( Q \) is constant and the cross-sectional area \( A \) decreases, the velocity \( V \) must increase to maintain the relationship \( Q = V \times A \). This is derived from rearranging the equation to \( V = \frac{Q}{A} \).

Conclusion of Stream Velocity Behavior

Since the area \( A \) decreases and discharge \( Q \) remains constant, the velocity \( V \) must increase. Thus, as water flows under the bridge where the stream channel narrows, the water velocity will increase.

Key concepts

Stream Flow

Stream flow refers to the movement of water within a defined channel, such as a river or a stream. It is essential for understanding how water travels through different landscapes. The primary factors influencing stream flow are the volume of water and the channel's shape and size. When we talk about stream flow, we're essentially observing how water is carried from one place to another.
To better grasp the concept:

• Stream flow is driven by the flow of water over land and its path through the channel.
• It affects erosion, sediment transport, and ecological habitat.
• Changes in stream flow can occur due to natural events or human activities.

In summary, understanding stream flow is crucial for managing water resources and protecting ecosystems around bodies of water.

Discharge Calculation

Discharge calculation is a vital part of fluid dynamics as it helps us quantify the amount of water traveling in a stream. Discharge, denoted by the symbol \( Q \), is defined as the volume of water passing a particular point in the stream per unit of time. It is expressed in units such as cubic meters per second (\( m^3/s \)). The fundamental formula for calculating discharge is:

• \( Q = V \times A \)

where \( V \) is the velocity of the stream and \( A \) is the cross-sectional area. This relationship shows:

• The discharge is directly proportional to both the flow velocity and the cross-sectional area.
• We can measure \( V \) using flow meters.
• Area \( A \) is calculated from channel width and depth at a given point.

With this understanding, calculating discharge helps predict how streams react under various conditions and is vital for managing water resources, designing hydraulic structures, and understanding environmental impacts.

Continuity Equation

The continuity equation is a fundamental principle used in physics and engineering, especially in fluid dynamics. It describes the conservation of mass in a stream flow, ensuring that mass is neither created nor destroyed as water moves through a system. In our context of fluid dynamics, the continuity equation is represented by \[Q = V \times A\]This states that the discharge \( Q \) remains constant in a closed system. Therefore, if one part of the stream narrows or widens:

• The velocity \( V \) inherently changes to accommodate the change in the cross-sectional area \( A \).
• If \( A \) decreases, \( V \) increases as long as \( Q \) remains constant.
• This also implies that if \( A \) increases, then \( V \) decreases for a constant discharge \( Q \).

The continuity equation helps explain many phenomena in fluid dynamics, such as why the water velocity increases when a stream narrows. It ensures that we understand and predict how fluids will behave in different scenarios, which is crucial for numerous engineering applications.