上游充通大学 Shanghai Jiao Tong University 第三章 流体动力学
Shanghai Jiao Tong University 第三章 流体动力学
上游充通大睾 3.1流体动力学的两个研究途径 Shanghai Jiao Tong University 流体动力学的两个研究途径：系统途径和控制体途径。 (I)系统途径(System approach):或称 为物质体(material volume)途径。对 由确定的流体质点所组成的流体团系 统的动力特性进行研究。Follow the System fluid as it moves and deforms; no mass crosses the I boundary。与 Lagrange法对应。 RTT (2)控制体途径(Control volume Control approach):对一个固定空间控制体的 volume 流体动力特性进行研究。Consider the changes in a certain fixed volume; mass can cross the boundary。与 Euler法对应 雷诺输运定理(Reynolds Transport Theorem (RTF)
Shanghai Jiao Tong University 3.1 流体动力学的两个研究途径 流体动力学的两个研究途径：系统途径和控制体途径。 (1) 系统途径(System approach): 或称 为物质体(material volume)途径。对 由确定的流体质点所组成的流体团系 统的动力特性进行研究。 Follow the fluid as it moves and deforms; no mass crosses the boundary 。 与 Lagrange法对应。 (2) 控制体途径 (Control volume approach): 对一个固定空间控制体的 流体动力特性进行研究。Consider the changes in a certain fixed volume; mass can cross the boundary 。 与 Euler法对应。 雷诺输运定理(Reynolds Transport Theorem (RTT))
上降充通大￥ 3.1流体动力学的两个研究途径 Shanghai Jiao Tong University 系统(system:由确定的流体质点组成的流体团。 物质体积Material Volume):由 G(x,y,z,t)dv 系统的流体团构成的体积(a ) volume that contains the same V() fluid as it moves and deforms following the motion of the fluid.) G(x,y,z,1) 物质表面Material Surface):物 质体积的封闭表面(enclosing surface of a material volume;by definition no fluid particles can cross it.)
Shanghai Jiao Tong University 3.1 流体动力学的两个研究途径 系统(system)：由确定的流体质点组成的流体团。 物质体积(Material Volume): 由 系统的流体团构成的体积(a volume that contains the same fluid as it moves and deforms following the motion of the fluid.) 物质表面(Material Surface): 物 质体积的封闭表面(enclosing surface of a material volume; by definition no fluid particles can cross it.)
上浒充通大￥ 3.1雷诺输运定理 Shanghai Jiao Tong University 控制体(control volume):一个空间体积。 Mass 控制体积(control Volume):由一个 Mass entering leaving 固定空间构成的体积(a volume of fluid in a flow field,usually fixed in space,to be occupied by different Control volume fluid particles at different times.) n outward normal 控制表面(control Surface):控制体 积的封闭表面(imaginary or Mass leaving physical enclosing surface of a control volume,fluid particles can Bnet=Bou-Bn=pbv.nidA cross it)
Shanghai Jiao Tong University 3.1 雷诺输运定理 控制体(control volume)：一个空间体积。 控制体积(control Volume): 由一个 固定空间构成的体积(a volume of fluid in a flow field, usually fixed in space, to be occupied by different fluid particles at different times.) 控制表面( control Surface): 控制体 积的封闭表面(imaginary or physical enclosing surface of a control volume, fluid particles can cross it)
h 上游充通大睾 3.1雷诺输运定理 Shanghai Jiao Tong University System Fixed control Fixed control volume volume occupies①and2 System at time t occupies volumes (1 and(2 2 3 System at time t+△t occupies volumes 2 and(3
Shanghai Jiao Tong University 3.1 雷诺输运定理
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