Chapter3 Integral Relations(积分关系式) for a control volume in One-dimensional steady Flows 31 Systems(体系) versus Control volumes(控制体) System: an arbitrary quantity of mass of fixed identity. Everything external to this system is denoted by the term surroundings, and the system is separated from ts surroundings by it's boundaries through which no mass across.( Lagrange拉格朗日) Control Volume(Cv: In the neighborhood of our product the fluid forms the environment whose effect on our product we wish to know. This specific region is called control volume, with open boundaries through which mass momentum and energy are allowed to across (Euler Efi) Fixed Cv, moving Cv, deforming Cv
3.1 Systems (体系) versus Control Volumes (控制体) System：an arbitrary quantity of mass of fixed identity. Everything external to this system is denoted by the term surroundings, and the system is separated from its surroundings by it‘s boundaries through which no mass across. (Lagrange 拉格朗日) Chapter 3 Integral Relations（积分关系式） for a Control Volume in One-dimensional Steady Flows Control Volume (CV): In the neighborhood of our product the fluid forms the environment whose effect on our product we wish to know. This specific region is called control volume, with open boundaries through which mass, momentum and energy are allowed to across. (Euler 欧拉) Fixed CV, moving CV, deforming CV
3.2 Basic Physical Laws of Fluid Mechanics All the laws of mechanics are written for a system, which state what happens when there is an interaction between the system and it's surroundings If m is the mass of the system Conservation of ma5质量守→ m= const or dm=0 Newtons second aw →F=m=mm= Angular dH momentum →M H=∑(×1)·m dt First law of do dw dE thermodynamic
3.2 Basic Physical Laws of Fluid Mechanics All the laws of mechanics are written for a system, which state what happens when there is an interaction between the system and it’s surroundings. If m is the mass of the system Conservation of mass(质量守恒) 0 dm m const or dt = = Newton’s second law F ma = dV m dt = ( ) d mV dt = Angular momentum dH M dt = H r V m = ( ) First law of thermodynamic dQ dW dE dt dt dt − =
It is rare that we wish to follow the ultimate path of a specific particle of fluid. Instead it is likely that the fluid forms the environment whose effect on our product we wish to know such as how an airplane is affected by the surrounding air, how a ship is affected by the surrounding water. This requires that the basic laws be rewritten to apply to a specific region in the neighbored of our product namely a control volume(CV. the boundary of the cv is called control surface ( cs) Basic Laws for system for CV 3. 3 The Reynolds Transport Theorem(RTT) 雷诺输运定理
It is rare that we wish to follow the ultimate path of a specific particle of fluid. Instead it is likely that the fluid forms the environment whose effect on our product we wish to know, such as how an airplane is affected by the surrounding air, how a ship is affected by the surrounding water. This requires that the basic laws be rewritten to apply to a specific region in the neighbored of our product namely a control volume ( CV). The boundary of the CV is called control surface(CS) Basic Laws for system for CV 3.3 The Reynolds Transport Theorem (RTT) 雷诺输运定理
1122 is CV 112>2 is system which occupies the cv at instant t 2 tt+at dp: any property of fluid(m, mv,H,e) dΦ B dm The amount of per unit mass The total amount of g in the cv is db=「B Cv
1122 is CV . 1*1*2*2* is system which occupies the CV at instant t . d dm = :The amount of per unit mass CV cv = d cv = dm The total amount of in the CV is : t+dt t+dt t t s : any property of fluid ( , , , ) m mV H E
(④cp) [Φcp(t+l)-Φc(t) t+at 2 tt+at s (t+dt)-(dq)out +(do)in]-aps(t dig(t+dt)-ops(t]-[(da )out -(dd )in] dt dΦs1 [dop) out-(ddp)in dt dt dΦde1 +,[(d)am-(d)m dt dt
1 [ ( ) ( )] CV CV t dt t dt = + − 1 1 [ ( ) ( ) ( ) ] ( ) out in s s t dt d d t dt dt = + − + − 1 1 [ ( ) ( )] [( ) ( ) ] s out in s t dt t d d dt dt = + − − − 1 [( ) ( ) ] s out in d d d dt dt − = − 1 [( ) ( ) ] s cv out in d d d d dt dt dt = + − ( ) CV d dt = t+dt t+dt t t s