工程流体力学第四章2

工程流体力学分析,ansys

Chapter 4 Differential Relations For Viscous Flow4.1 Preliminary Remarks* Two ways in analyzing fluid motion (1) Seeking an estimate of gross effects over a finite region or control volume.

Integral(2) Seeking the point-by-point details of a flow pattern by analyzing an infinitesimal region of the flow.

Differential

工程流体力学分析,ansys

* Viscous flow Viscosity is inherent nature of real fluid. Strain(剪切) is very strong in internal flow. * Two forms of flow Turbulent(湍，紊) flow, laminar(层)flow Turbulent Flow VS. Laminar Flow TransitionReynolds tank

UL Reynolds Re 惯性力/粘性力 number

Osbrone Reynolds 2

工程流体力学分析,ansys

4.2 The Acceleration Field of a Fluid V (r , t ) iu( x, y, z, t ) jv( x, y, z, t ) kw( x, y, z, t ) dV du dv dw a i j k dt dt dt dt

du u u x u y u z dt t x t y t z t

u u u u u v w t x y zLocal acceleration 0 unsteady t

Nonlinear terms

Convective acceleration

nonuniform

0 xi3

工程流体力学分析,ansys

V a (V )V tIn the like manner Any property Φ

, T ... D (V ) Dt t

D (V ) Dt t

Substantial (Material) derivative 随体(物质、全)导数

工程流体力学分析,ansys

Example 2 Given V 3ti xzj ty k .

Find the acceleration of a particle.u 3t, v xz, w ty 2du u u u u u v w 3 dt t x y z

dv v v v v u v w 3tz ty 2 x dt t x y z

dw w w w w y 2 2tyxz u v w dt t x y z5

工程流体力学分析,ansys

4.3 Differential Equation of Mass ConservationX inlet (mass flow)y ( u) dx]dydz x

udydz u mass flux per unit areaX outlet udydz

dy

[ u

x

( u) dz [ u dx]dydz z x dx ( u) X flow out dxdydz Infinitesimal fixed CV x ( v) ( w) dxdydz In the like manner dxdydz y z Flow out off the CV ( V )dxdydz6

工程流体力学分析,ansys

dxdydz Loss of mass in the CV t ( u ) ( v) ( w) dxdydz dxdydz dxdydz dxdydz x y z t ( u ) ( v) ( w) dxdydz dxdydz dxdydz dxdydz 0 t x y z

( u ) ( v) ( w) 0 t x y z u v w V 0 t x y z

( V ) 0 t

d V 0 dt7

工程流体力学分析,ansys

For steady flow ( V ) 0 For incompressible flow V 0Example 1 Under what conditions does the velocity field V (a1 x b1 y c1 z )i (a2 x b2 y c2 z ) j (a3 x b3 y c3 z )k

represents an incompressible flow which conserves mass? ( where ai , bi , ci const )8

工程流体力学分析,ansys

Solution

u

a1x b1 y c1z v a2 x b2 y c2 zContinuity for incompressible flow u v w 0 x y z

w a3 x b3 y c3 za1 b2 c3 0

Example 2 An incompressible velocity field: u=a(x2-y2),w=b, a,b are const,what v=?Solution

u v w 0 x y z

v 2ax 0 y

v 2ax y

v 2axy f ( x, z, t )

An arbitrary function of x,z,t

工程流体力学分析,ansys

Assignment: P264: P4.1(a), P4.2, P4.4 ,P4.9(a)

工程流体力学分析,ansys

4.4 Differential Equation of Linear Momentum Newton’s second law dV F ma dxdydz dt dV dxdydz Fb Fs dtdx

dy

Fb R dxdydz

dz

( Xi Yj Zk ) dxdydz

Elemental volume

What are the surface forces Fs on the elemental volume?11

工程流体力学分析,ansys

Surface force on an elemental volume:

for the two surfaces x Px dydz Px Surface stress

Px ( Px dx)dydz x

Pxdx

Px Px dx x dy

dz

Vector Sum

Px dxdydz x

for the two surfaces y, zNet Surface Force:

dxdydz y Px Py Pz Fs ( )dxdydz x y z

Py

Pz dxdydz z

工程流体力学分析,ansys

It is not these stresses but their gradient, which cause a net force on the differential volume. Px Py Pz dV dxdydz Rdxdydz ( )dxdydz dt x y z dV 1 Pi R [ ] dt xi

Momentum equation

Px xxi xy j xz kIn the like manner Py yxi yy j yz k Pz zxi zy j zz k

xy

xz

xx

Px

工程流体力学分析,ansys

xx xy xz yx yy yz zx zy zz

ij

Tensor 张量

ij ji i j

6

xx yx zx xy yy zy Fs [( )i ( )j x y z x y z xz yz zz ( )k ]dxdydz x y z

du 1 xx yx zx X ( ) dt x y z

dv dw ... ... dt dt14

工程流体力学分析,ansys

dui 1 ji Xi ( ) Momentum equation(角标表示法) dt x j

du w dy

u ui xy ij ~ y x j

Constitutive Relation 本构

Newton’s Law (广义牛顿内摩擦定律) 2 ij 2 Sij ( V ) ij P ij 3

1 ui u j Sij ( ) 2 x j xi

ij

1

i ji j15

0

工程流体力学分析,ansys

2 ij 2 Sij ( V ) ij P ij 3 u 2 xx 2 ( V ) P x 3 v 2 yy 2 ( V ) P y 3

u v xy ( ) y x u w xz ( ) z x w v yz ( ) y z

w 2 zz 2 ( V ) P z 3

Newton flu

id, linear fluid (牛顿流体，线性流体)

Substitute Newton’s Constitutive Relation into ME16