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5.2 Velocity and Acceleration Fields

3 min readLast Updated on July 19, 2024

Velocity and acceleration fields are crucial tools for understanding fluid motion. They assign vectors to each point in a fluid, describing the speed, direction, and rate of change of fluid particles. These fields provide a complete picture of fluid behavior in space and time.

Partial derivatives and vector plots help visualize and analyze fluid motion. Streamlines show fluid paths, while material derivatives connect fixed-point and particle-following descriptions. These concepts are essential for grasping how fluids move and change in various scenarios.

Velocity and Acceleration Fields

Velocity and acceleration fields

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  • Velocity field assigns a velocity vector to each point in a fluid domain (water flowing in a pipe)
    • Represents instantaneous velocity of fluid particles at each point
    • Denoted as V(r,t)\vec{V}(\vec{r},t), where r\vec{r} is position vector and tt is time
  • Acceleration field assigns an acceleration vector to each point in a fluid domain (air accelerating over an airplane wing)
    • Represents instantaneous acceleration of fluid particles at each point
    • Denoted as a(r,t)\vec{a}(\vec{r},t), where r\vec{r} is position vector and tt is time
  • Interpretation of velocity and acceleration fields
    • Velocity field describes speed and direction of fluid motion at each point (wind velocity in a hurricane)
    • Acceleration field describes rate of change of velocity at each point (acceleration of water in a contracting pipe)
    • Both fields provide complete description of fluid motion in space and time

Partial derivatives for fluid components

  • Velocity components in Cartesian coordinates calculated using partial derivatives
    • u=xtu = \frac{\partial x}{\partial t}, v=ytv = \frac{\partial y}{\partial t}, w=ztw = \frac{\partial z}{\partial t}
    • V=ui^+vj^+wk^\vec{V} = u\hat{i} + v\hat{j} + w\hat{k} (velocity vector in 3D space)
  • Acceleration components in Cartesian coordinates calculated using partial derivatives
    • ax=ut+uux+vuy+wuza_x = \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z}
    • ay=vt+uvx+vvy+wvza_y = \frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z}
    • az=wt+uwx+vwy+wwza_z = \frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z}
    • a=axi^+ayj^+azk^\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k} (acceleration vector in 3D space)
  • Partial derivatives calculate rate of change of velocity and acceleration components with respect to space and time (velocity gradient in a shear flow)

Vector plots of fluid fields

  • Vector plots represent magnitude and direction of velocity or acceleration vectors at each point in fluid domain (wind velocity map)
    • Arrow length indicates magnitude, arrow direction indicates direction of vector
  • Streamlines are curves tangent to velocity vectors at each point (smoke trails in a wind tunnel)
    • Represent instantaneous path of fluid particles
    • Closely spaced streamlines indicate high velocity, widely spaced streamlines indicate low velocity (streamlines around an airfoil)
  • Analysis of vector plots and streamlines
    • Identify regions of high and low velocity or acceleration (stagnation points in a flow)
    • Locate stagnation points (zero velocity) and vortices (circular motion) (vortex shedding behind a cylinder)
    • Determine presence of shear and normal stresses in fluid (shear stress in a boundary layer)

Material derivative in fluid descriptions

  • Eulerian description focuses on fluid properties at fixed points in space (pressure field in a room)
    • Velocity and acceleration fields described as functions of position and time
  • Lagrangian description follows individual fluid particles as they move through space and time (trajectory of a pollutant particle in a river)
    • Tracks position, velocity, and acceleration of each particle
  • Material derivative (total derivative) relates Eulerian and Lagrangian descriptions
    • Represents rate of change of a fluid property following a fluid particle
    • For a scalar property ϕ\phi: DϕDt=ϕt+uϕx+vϕy+wϕz\frac{D\phi}{Dt} = \frac{\partial \phi}{\partial t} + u\frac{\partial \phi}{\partial x} + v\frac{\partial \phi}{\partial y} + w\frac{\partial \phi}{\partial z} (temperature change in a moving fluid)
    • For velocity: DVDt=Vt+(V)V\frac{D\vec{V}}{Dt} = \frac{\partial \vec{V}}{\partial t} + (\vec{V} \cdot \nabla)\vec{V} (acceleration of a fluid particle)
  • Material derivative accounts for both local (Eulerian) and convective (Lagrangian) changes in fluid properties (heat transfer in a flowing fluid)
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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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