Introduction to Tensor Index Notation in Fluid Mechanics, Lecture notes of Fluid Mechanics

Download Introduction to Tensor Index Notation in Fluid Mechanics and more Lecture notes Fluid Mechanics in PDF only on Docsity!

Introduction to Tensor Index Notation

AAE 511 Introduction to Fluid Mechanics

Jonathan Poggie Fall 2020

School of Aeronautics and Astronautics Purdue University

Copyright

This presentation contains material copyrighted by the instructor and by other authors. These materials are intended only for personal educational use in the context of this class.

About this Class

The class is organized in three main sections

1. Fundamentals of fluid mechanics
2. Potential flow
3. Incompressible, viscous flow

Fundamentals of fluid mechanics

• Tensors
• Kinematics
• Conservation equations
• Constitutive relations
• Vorticity dynamics

Potential flow

• Complex analysis
• Potential flow
• Method of images
• Conformal mapping
• Schwarz-Christoffel transformation
• Axisymmetric and 3D potential flow

Introduction

See references for more information

For reference see Appendix A of Currie (2013), Chapter 3 of Panton (1984), or Aris (1962)

A vector has magnitude and direction

A vector has magnitude and direction, for example position vector:

x = (x 1 , x 2 , x 3 )

A vector field defines a vector at each location in space, for example velocity:

v (x) = (v 1 (x 1 , x 2 , x 3 ), v 2 (x 1 , x 2 , x 3 ), v 3 (x 1 , x 2 , x 3 ))

Form Cartesian coordinates with orthonormal basis vectors

Orthonormal basis vectors: e 1 , e 2 , e 3 Normal: e 1 ·e 1 = 1, etc Orthogonal: e 1 ·e 2 = 0, etc A vector can be represented as:

a = (a 1 , a 2 , a 3 ) a = a 1 e 1 + a 2 e 2 + a 3 e 3 ak = (a)k = a·ek , for k = 1, 2 , 3

𝑥!

𝑥"

𝑥#

𝒂

𝑎"

𝑎!

𝑎# 𝒆"

𝒆! 𝒆#

Non-Cartesian coordinates are sometimes convenient

• We will sometimes use orthogonal curvilinear coordinates
• Examples are cylindrical and spherical coordinates
• See Currie (2013), Appendices A, C

The dot product is a scalar

a·b = a 1 b 1 + a 2 b 2 + a 3 b 3 Perpendicular vectors have zero dot product: a·b = 0 if a ⊥ b The dot product can give the vector magnitude:

a^2 = a·a a = |a| =

a·a

A tensor generalizes scalar and vector

• A tensor has certain linear behavior under rotation of coordinate axes
• A tensor of rank r has 3r^ components in three dimensions
• A scalar is a rank zero tensor (e.g., pressure)
• A vector is a rank one tensor (e.g., velocity)
• A dyadic (or dyad) is a rank two tensor (e.g., stress)

Number of tensor components grows rapidly with rank

Name Example Rank Components

Scalar p 0 1 Vector vi 1 3 Dyadic τij 2 9 ijk 3 27 γijk` 4 81