高等流体力学第一篇introduction

高等流体力学Advanced Fluid Mechanics

主讲：余永亮

中国科学院大学

工程科学学院，北京100049

Chapter 2 Viscous Fluid Motion

§2.1 Introduction

•Governing Equations

•Conditions of the definite solutions of Navier-Stokes

Equations

•Mathematical Properties of Navier-Stokes Equations •Similarity Parameters

1. Governing Equations (1) Continuity Equation

(2) Dynamics(Kinetics) Equation

Constitutive Relation:

For incompressible flow,

Navier-Stokes Equation (incompressible)

Internal Energy

1. Governing Equations

(3) Energy Equation

Fourier ’

s Law Viscous dissipation function

Total kinetic energy (incompressible and uniform fluid)

The change rate of the total kinetic energy

Total kinetic energy (incompressible and uniform fluid)

The change rate of the total kinetic energy

=0, for an isolated system

Total kinetic energy (incompressible and uniform fluid)The change rate of the total kinetic energy

The viscosity coefficient is always positive

The second law of

thermodynamics

1. Governing Equations (*) State Equation

This set of equations is complete！

2. Conditions of the definite solution of N-S-E •Boundary Condition + Initial Condition

•Physical law

•

Mathematical properties

For Euler equation

At the solid boundary

(1) Solid Boundary

Suppose : No mass exchange at the solid surface Boundary Condition: Non-slip condition(Adhesive Con.)

the boundary condition can not be proved

the boundary conditions are conditional!

For porous surfaces, there is mass exchange

(2) Free Surface

(2) Free Surface

Kinematic condition:

(2) Free Surface

Dynamic condition:

I. No surface Tension

II. With surface Tension

(3) Energy Condition Notice: For viscous flows, we don’t

recognize there exists discontinuity

in the flow field.

•PDEs (partial differential equations) with 2 independent variables

All coefficients are sufficiently smooth

e.g. 1D wave

Initial value:

Exact solution:

characteristic equation

Strictly Hyperbolic Equations(狭义双曲型方程组): there are N

different real roots of this equation at every point (real eigenvalues)

Elliptic Equations(椭圆型方程组): there is no real roots of this

equation at every point.

•The standard form of the second-order partial differential equations

Where A,B,C,D are the function of

Boundary-value problem

Initial and Boundary-value problem

Initial and Boundary-value problem

Initial-value problem