流体力学-英文课件第4章

1、1Chapter 4 Differential Relations ForViscous Flow(3D viscous flow)24.1 Preliminary Remarks* Two ways in analyzing fluid motionSeeking 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 r

2、egion of the flow. Differential3Laminar Flow VS. Turbulent Flow* Two forms of flowLaminar flow （层流）, Turbulent flow （湍流） * Viscous flowViscosity is inherent nature of real fluid.TransitionReynolds numberOsbrone ReynoldsReynolds tank44.2 The Acceleration Field of a FluidLocal accelerationunsteadyConv

3、ective accelerationnonuniformNonlinear terms5In the like mannerAny property Substantial Derivative随体（物质、全）导数6ExampleGiven . Find the acceleration of a particle.7X inlet (mass flow)X outletdxyzxdzdyInfinitesimal fixed CVX flow out4.3 Differential Equation of Mass ConservationIn the like manner Flow o

4、ut of the CV8Loss of mass in the CV9For steady flowFor pressible flowExample 1Under what conditions does the velocity fieldrepresents an pressible flow which conserves mass? ( where ) 10Solution:Continuity for pressible flowExample 2An pressible velocity field:u=a(x2-y2),w=b, a,b are const,what v=?S

5、olution:An arbitrary function of x,z,t11Assignment:Page264: P4.1(a), P4.2, P4.4 ,P4.9(a)12Newtons second law4.4 Differential Equation of Linear MomentumdxdzdyElemental volumeWhat are the surface forces Fs on the elemental volume?13Surface force on an elemental volume:dxdzdyVector SumSurface stressNe

6、t Surface Force:14Momentum equationIn the like manner15Tensor 张量16Constitutive Relation 本构方程Newtons Law (广义牛顿内摩擦定律) Momentum equation(角标表示法)17Substitute Newtons Constitutive Relation into MENewton fluid, linear fluid (牛顿流体，线性流体）18N-S Equation19For pressible flowFor inviscid flowFor pressible flow202

7、14.5 The Differential Equation of EnergyInfinitesimal fluid elementdxdzdyThe first thermodynamic law22(1) Thermal conductivity (2) othersX: Heat flowdxdydzAccording to Fouriers Law:23质量力做功和表面力做功Body force24Surface forcedxdydzX:X: Net powerY,Z25Net power by Fsleft26单位体积内能变化率热传导等传热变形时表面应力做功27变型时表面应力做功

8、压力做膨胀功粘性耗散 0根据热力学公式，熵s、焓h和压强p、密度的关系为：28已知 D = CvDT, Dh = CpDT29Summary of the Equations303132equationsunknown variablescontinuity1r ,u,v,wmomentum3p, r ,u,v,wenergy1p, r, u,v,w,Tperfect gas1p,r,TTo Solve A Flow334.6 Initial (初始)and Boundary（边界）Conditions for the Basic EquationsInitial Conditions：t =

9、 t0 :Boundary Conditions：(No slip) VelocityWallIf the wall is stationaryTemperature34Due to the highly complex of the N-S equations, only a few particular solutions were found up to now. For most problems, the equations must be solved numerically, which is a brand new course called CFD (Computationa

10、l Fluid Dynamics)Flow pass a cylinder An experiment resultA computation resultSolving the N-S equations numerically35xyohhUFlow between two parallel walls, Steady, pressible, neglect body force, 2-D Continuity:Momentum:4.7 Exact solutions of N-S Equations36= constantIntegrate relative to yBoundary c

11、ondition:xyohhU37Apply the boundary conditionxyoh-hu(y)Uxyoh-hWhen U=0Poiseuille flowWhenSimple Couette flow38WhenWhenConsider a special caseGeneral case:yxoUxyoUxoyUCouette flow39Q=0:yxoVolume flow rateQ=u*dy*140The wall shear stresses414.8 Dynamical Similarity & NondimensionalizationSimilarity D =

12、 5cm D = 10cm42N-S equation, 2-D, steady, no body force, pressibleUse U,L as reference velocity and lengthDimensionless quantitiesx direction:4.8.1 Nondimensionalization of N-S Equation4344For steady, pressible, no body force flow, if two geometrically similar flow fields has same Reynolds number, then they have similar flow structure when same boundary conditions are provided.Why Reynolds number?45Inertia force / viscous force两个铁球同时落地？46Flow pass a cylinder D = 5cm D = 10cmFlow pass a square Re = 50 Re = 10000The flow fields for two objects of the same shape but different size