5.2 Velocity and Acceleration Fields

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 $\vec{V}(\vec{r},t)$, where $\vec{r}$ is position vector and $t$ 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 $\vec{a}(\vec{r},t)$, where $\vec{r}$ is position vector and $t$ 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
  • $u = \frac{\partial x}{\partial t}$, $v = \frac{\partial y}{\partial t}$, $w = \frac{\partial z}{\partial t}$
  • $\vec{V} = u\hat{i} + v\hat{j} + w\hat{k}$ (velocity vector in 3D space)
• Acceleration components in Cartesian coordinates calculated using partial derivatives
  • $a_x = \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z}$
  • $a_y = \frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z}$
  • $a_z = \frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z}$
  • $\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$ (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 $\phi$: $\frac{D\phi}{Dt} = \frac{\partial \phi}{\partial t} + u\frac{\partial \phi}{\partial x} + v\frac{\partial \phi}{\partial y} + w\frac{\partial \phi}{\partial z}$ (temperature change in a moving fluid)
  • For velocity: $\frac{D\vec{V}}{Dt} = \frac{\partial \vec{V}}{\partial t} + (\vec{V} \cdot \nabla)\vec{V}$ (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)