Physlib.FluidDynamics.FluidFlow.Kinematics

Kinematic quantities for fluid flows

i. Overview

This module defines basic kinematic scalar quantities associated to `FluidFlow`.

ii. Key results

`FluidFlow.materialDerivative` : The material derivative along a fluid velocity field.
`FluidFlow.specificKineticEnergy` : The specific kinetic energy `|u|^2 / 2`.

iii. Table of contents

A. Flow-derived scalar quantities

iv. References

A. Flow-derived scalar quantities

2 declarations

definition

Material derivative $D_t f = \partial_t f + \mathbf{u} \cdot \nabla f$ of a scalar field

#materialDerivative

Given a fluid flow in $d$ -dimensional space with velocity field $\mathbf{u}(t, \mathbf{x})$ and a time-dependent scalar field $f(t, \mathbf{x})$ , the material derivative $D_t f$ is a scalar field defined by: $(D_t f)(t, \mathbf{x}) = \frac{\partial f}{\partial t}(t, \mathbf{x}) + \mathbf{u}(t, \mathbf{x}) \cdot \nabla f(t, \mathbf{x})$ where $\frac{\partial f}{\partial t}$ is the partial derivative of the scalar field with respect to time $t$ , $\nabla f$ is the spatial gradient of the field at a fixed time, and $\cdot$ denotes the standard Euclidean inner product.

definition

Specific kinetic energy $\frac{1}{2} |\mathbf{u}|^2$ of a fluid flow

#specificKineticEnergy

Given a fluid flow in $d$ -dimensional space with velocity field $\mathbf{u}(t, \mathbf{x})$ , the specific kinetic energy is the scalar field defined by mapping each point in time $t$ and space $\mathbf{x}$ to the value $\frac{1}{2} \|\mathbf{u}(t, \mathbf{x})\|^2$ , where $\|\cdot\|$ denotes the standard Euclidean norm.