Correlation between System (Lagrangian) concept Control-volume (Eulerian) concept for comprehensive understanding of fluid motion?

Transcription

1 The Reynolds Tanspot Theoem Coelation between System (Lagangian) concept Contol-volume (Euleian) concept fo compehensive undestanding of fluid motion? Reynolds Tanspot Theoem Let s set a fundamental equation of physical paametes B mb e.g. a) If b) If whee B: Fluid popety which is popotional to amount of mass (Extensive popety) b: B pe unit mass (Independent to the mass) (Intensive popety) B mv (Linea momentum): Extensive popety then, b V (Velocity): Intensive popety 1 B mv 2 (Kinetic enegy): Extensive popety then, b V : Intensive popety 2

2 i. B of a system B sys at a given instant, B sys lim bi ( ρiδvi ) δv 0 i sys ρ bdv δ m i fo i th fluid paticle in the system And Time ate of change of B sys, ( ) db sys d sys ρbdv dt dt whee δ Vi : Volume of i th fluid paticle ii. B of fluid in a contol volume B and B lim bi ( ρiδvi ) δv 0 db d ( ρbdv ) dt dt i ρ bdv Only diffeence fom B of a system Relationship between db sys dt and db : Reynolds Tanspot Theoem dt

3 Deivation of the Reynolds Tanspot Theoem Conside 1-D flow though a fixed contol volume shown Fixed contol suface at t (coincide with a system bounday) System bounday at t + a) At time t, Contol volume (CV) & System (SYS): Coincide b) At t + (afte δ t ), CV: fixed & SYS: Move slightly Fluid paticles at section (1): Move a distance Fluid paticles at section (2): Move a distance I : Volume of Inflow (enteing CV) II : Volume of Outflow (leaving CV) dl dl 1 2 V 1 V 2 That is, SYS (at time t) CV SYS (at time t + ) CV I + II O if B: Extensive fluid popety, then B sys (t) B (t) (at time t) B t + δ t) B ( t + ) B ( t + ) + B ( t + ) (at time t + ) sys ( I II

4 Then, Time ate of change in B can be; δ Bsys Bsys ( t + ) Bsys ( t) B (t), at time t B B In the limit δ t 0, ( I II t + δ t) B ( t + ) + B ( t + ) B ( t) sys t + δ t) B( t) BI ( t + ) B ( t + ) + ( II Left-side: δ B sys 1 st tem on Right-side: DB sys lim 0 B (accoding to Lagangian Concept) ( t + ) B ( t) B ρbdv 2 nd B tem on Right-side: I( t t) B& + δ in lim ρ1av 1 1b1 (4.13) 0 because B ( t + t) ( ρ δv b ρ AV b I δ 1 1) whee A 1: Aea at section (1) V 1: Velocity at section (1) 3 d B tem on Right-side: II( t t) B& + δ lim ρ2a2v 2b2 (4.12) 0 because B II δ 2 2) ( t + t) ( ρ δv b ρ A V b

5 Relationship between the time ate of change of B sys and B DB B B sys + B& B& in + ρ2a2v 2b2 ρ1av 1 1b1 : Special vesion of Reynolds tanspot theoem - Fixed CV with one inlet and one let - Velocity nomal to Sec. (1) and (2) Geneal expession of Reynolds Tanspot Theoem Conside a geneal flow shown At time t, CV & SYS: Coincide At time t +, CV: Fixed & SYS: Move slightly DBsys B + B& Still valid, but B& B& in & B& in: Diffeent What ae B& & B& in?

6 1) B& : Net flowate of B leaving CV (Outflow) acoss the contol suface between II and CV ( ) B acoss the aea element δ A on δ B bρδv bρ( V cosθ) δa whee δ V (Fluid volume leaving CV acoss δa Then, the time ate of B acoss δ A δ δa δl cos θδa ( V cosθ ) δa l n δb& lim 0 ρbδv ( ρbv lim 0 cosθ) δa ρbv cosθδa By integating ove the entie, B& db& ρ bv cosθda ρbv nda ˆ

7 2) B& : Net flowate of B enteing CV (Inflow) acoss the contol suface between I and CV ( ) in By the simila manne, B& in ρ bv cosθda in in (because π 3π < θ < ) 2 2 ρbv nda ˆ Finally, Net flowate of B acoss the entie ( + ) B& B& in ρ bv nda ˆ ( in ρ bv nˆ da in ρbv nˆ da) DB sys B + ρ bv nda ˆ bdv + ρ ρbv nda ˆ : Geneal expession of Reynolds Tanspot Theoem

8 PHYSICAL INTERPRETATION DB sys : Time ate of change of an extensive B of a system Lagangian concept ρ bdv : Time ate of change of B within a contol volume Euleian concept ρ bv nˆ da: Net flowate of B acoss the entie contol suface Coelation tem Motion of a fluid c.f. Compaison with the definition of Mateial Deivative () () () ( ) ( ) ( ) D + u + v + w + ( V )( ) x y z D() () ( V ) : Time ate of change of a popety of fluid paticle Lagangian concept : Time ate of change of a popety at a local space Euleian concept: Unsteady effect : Change of a popety due to the fluid motion Coelation tem Convective effect Reynolds Tanspot Theoem Tansfe fom Lagangian viewpoint to Euleian one (Finite size)

9 Special cases DBsys 1. Steady Effects. ρ bdv + bv nˆ da ρ Any change in popety B of a system Net diffeence in flowates B& enteing CV and leaving CV CV 2. Unsteady Effects. ρ bdv 0 Any change in popety B of a system Change in B within CV + Net diffeence in flowates B& enteing and leaving CV e.g. Fo 1-D flow V V 0 ( t) iˆ ρ Constant Choose B mv (Momentum), and thus b B / m V V t ) i ˆ 0 ( ρ bv nda ˆ ρ( V0iˆ) V nda ˆ ( V iˆ ) V da ( V iˆ ) V da ( V iˆ o ( ) + ρ ( ) )( V cos90 ) da ρ (1) 0 0 (2) ˆ 2 V ˆ 0 Ai + ρv0 A 0 ρ side ρ i (Inflow of B Outflow of B) 0 0 DB sys CV ρ bdv : No convective effect

10 Reynolds Tanspot Theoem fo a moving contol volume DB sys ρ bdv + bv nda ˆ ρ : Valid fo a stationay CV In case of moving contol volume as shown, Conside a constant velocity of CV V Reynolds tanspot theoem : Relation between a system and CV, (Neglect the suounding) Velocity of a system: Defined w..t. the motion of CV Relative velocity of a system: W V VCV whee V : Absolute velocity of a system Finally, DB sys ρ bdv + ρbw nˆ da : Valid fo a stationay o moving CV with constant V