h 上游充通大睾 3.1雷诺输运定理 Shanghai Jiao Tong University dA Control surface,c.s. dA dA n System and control System at volume identical time t+△r at time t Control volume at time t+△t (a) (b)
Shanghai Jiao Tong University 3.1 雷诺输运定理
上游充通大睾 3.2雷诺输运定理 Shanghai Jiao Tong University 雷诺输运定理Reynolds Transport Theorem) Control volume at time t+At (CV remains fixed in time) 任何一个物理量G都满足下列物质体积与控 System(material volume) and control volume at time t 制体积的关系式： (shaded region) System at time t+△t (hatched region) GdV GdV GV.ndA + rate of change of the local rate of change of the net out-flux of the property within property within the fixed property across the the material volume control volume that happens entire control surface to coincide with the material volume at that instant p=density of fluid 2) G=an intensive property of fluid MV material volume that happens to coincide with CV at time t Inflow during△t CI=control volume (fixed in space) Outflow during△t CS control surface n unit outward normal to CS At time t:Sys =CV At time t+△t:Sys=CV-I+Ⅱ
Shanghai Jiao Tong University 3.2 雷诺输运定理 雷诺输运定理(Reynolds Transport Theorem) rate of change of the local rate of change of the property within property within the fixed the material volume control volume that happens to coincide with the material volume MV CV d dV dV dt t G G ∂ = ∂ ∫∫∫ ∫∫∫ 1442443 net out-flux of the property across the entire control surface at that instant CS + ⋅ G dA ∫∫ V n 1442443 1442443 materi dens al v ity of f olume th luid an intensi at happens to ve property o coincide f with at time fluid MV CV t G ρ = = = control volume (fixed in space) control surface unit outward normal to CV CS CS = = n = 任何一个物理量G都满足下列物质体积与控 制体积的关系式：
上泽充通大￥ 3.2雷诺输运定理 Shanghai Jiao Tong University 对于物质体，在t后为： ∬c(kr -cur-c+a+or +d MV(t+dt) MV(t+dt) 在d内物质体的变化为：川()=()+旷()=()+「()vnd MV (t+dt) CS 因此有： o(.)rs.w 删去二阶小量 -则c小pr+[cx器a dm o(wc(.v.nuis Control volume(CV) 由上式可以得到RTT: or-or.-cw-容r+ovas Control surface(CS)
Shanghai Jiao Tong University 3.2 雷诺输运定理 ( ) ( ) () 2 ( ) ( ) , , ,( ) + + + ⎡ ⎤ ⎡ ∂ ⎤ ⎢ ⎥ = + = ++ ⎢ ⎥ ⎣ ∂ ⎦ ⎣ ⎦ ∫∫∫ ∫∫∫ ∫∫∫ MV MV t dt MV t dt t dt G G t dV G t dt dV G t dt O dt dV t xx x ( ) ( ) ( ) ( ) ( ) MV t d ( )t M V V CV CS dSdt + Δ =+=+ ⋅ ∫∫∫ ∫∫∫ ∫∫∫ ∫∫∫ ∫∫ V n ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , , , , + + ⎡ ⎤ ⎡ ⎤ ∂ ⎢ ⎥ = + ⎢ ⎥ ⎣ ⎦ ∂ ⎣ ⎦ ⎡ ⎤⎡ ⎤ ∂ ∂ = + + +⋅ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ∂ ∂ ⎡ ⎤ ∂ = ++⋅ ⎢ ⎥ ∂ ⎣ ⎦ ∫∫∫ ∫∫∫ ∫∫∫ ∫∫ ∫∫∫ ∫∫∫ ∫∫ MV MV t dt t dt CV CS CV CV CS G G t dV G t dt dV t G G G t dt dV G t dt dSdt t t G G t dV dV G t dS dt t x x x xV n x x V n MV MV t dt MV CV CS d G GdV GdV GdV dt dV G dS dt t + ⎪ ⎪ ⎧ ⎫ ⎡ ⎤ ∂ = − = +⋅ ⎨ ⎬ ⎢ ⎥ ∂ ⎪ ⎪ ⎩ ⎭ ⎣ ⎦ ∫∫∫ ∫∫∫ ∫∫∫ ∫∫∫ ∫∫ V n 对于物质体，在dt后为： 在dt内物质体的变化为： 因此有： 由上式可以得到RTT： 删去二阶小量
上游充通大学 3.2雷诺输运定理 Shanghai Jiao Tong University Lagrangian B Eulerian description description Control System analysis RTT volume analysis
Shanghai Jiao Tong University 3.2 雷诺输运定理
上游充通大学 3.3连续方程 Shanghai Jiao Tong University 连续方程(Continuity Equation):也称为质量守恒方程 (Conservation of Mass 在RTT方程中，如果物理量为质量，即G=P,可以得到： L.H.S. 乐∬nnr-（ms in)-0 (由MV的定义：在MV中总是包含相同的流体。) R.H.S. diar+八pdA =瓜g业+(o4 CI is stationary by Gauss theorem
Shanghai Jiao Tong University 3.3 连续方程 连续方程(Continuity Equation )：也称为质量守恒方程 (Conservation of Mass ) 在RTT方程中，如果物理量为质量，即 ，可以得到： L.H.S. ( ) mass in 0 ( MV MV ) MV d d dV MV dt dt ρ = = ∫∫∫ 由 的定义：在 中总是包含相同的流体。 ( ) by Gauss theorem is stationary R.H.S. = CV C C V CV C S V dV dA t dV dV t ρ ρ ρ ρ ∂ + ⋅ ∂ ∂ + ⋅ ∂ ∫∫∫ ∫∫ 14 ∫∫∫ ∫∫∫ 42443 1442443 V n ∇ V G = ρ