高等流体力学Advanced Fluid Mechanics主讲：余永亮中国科学院大学工程科学学院，北京100049Chapter 2 Viscous Fluid Motion§2.1 Introduction•Governing Equations•Conditions of the definite solutions of Navier-StokesEquations•Mathematical Properties of Navier-Stokes Equations •Similarity Parameters1. Governing Equations (1) Continuity Equation(2) Dynamics(Kinetics) EquationConstitutive Relation:For incompressible flow,Navier-Stokes Equation (incompressible)Internal Energy1. Governing Equations(3) Energy EquationFourier ’s Law Viscous dissipation functionTotal kinetic energy (incompressible and uniform fluid)The change rate of the total kinetic energyTotal kinetic energy (incompressible and uniform fluid)The change rate of the total kinetic energy=0, for an isolated systemTotal kinetic energy (incompressible and uniform fluid)The change rate of the total kinetic energyThe viscosity coefficient is always positiveThe second law ofthermodynamics1. Governing Equations (*) State EquationThis set of equations is complete！2. Conditions of the definite solution of N-S-E •Boundary Condition + Initial Condition•Physical law•Mathematical propertiesFor Euler equationAt the solid boundary(1) Solid BoundarySuppose : No mass exchange at the solid surface Boundary Condition: Non-slip condition(Adhesive Con.)the boundary condition can not be provedthe boundary conditions are conditional!For porous surfaces, there is mass exchange(2) Free Surface(2) Free SurfaceKinematic condition:(2) Free SurfaceDynamic condition:I. No surface TensionII. With surface Tension(3) Energy Condition Notice: For viscous flows, we don’trecognize there exists discontinuityin the flow field.•PDEs (partial differential equations) with 2 independent variablesAll coefficients are sufficiently smoothe.g. 1D waveInitial value:Exact solution:characteristic equationStrictly Hyperbolic Equations(狭义双曲型方程组): there are Ndifferent real roots of this equation at every point (real eigenvalues)Elliptic Equations(椭圆型方程组): there is no real roots of thisequation at every point.•The standard form of the second-order partial differential equationsWhere A,B,C,D are the function ofBoundary-value problemInitial and Boundary-value problemInitial and Boundary-value problemInitial-value problem•Three typical types of equationsPoisson Eqn. ---EllipticHeat Conduction Eqn. ---ParabolicWave Eqn. ---Hyperbolic•The simplified Navier-Stokes equations (No convective term, Uniform fluid, Mass force has a potential)àElliptic equationàParabolic equation Euler Equations for steady flow:Ma>1, à4 real rootsàHyperbolic EquationsMa=1, à2 group of real roots àParabolic-Hyperbolic EquationsMa<1,à1 pair of complex roots àElliptic-Hyperbolic EquationsEuler Equations for steady flow:Ma>1, à4 real rootsàHyperbolic EquationsMa=1, à2 group of real roots àParabolic-Hyperbolic Equations Ma<1,à1 pair of complex roots àElliptic-Hyperbolic Equations4. Similarity Parameters (1) Flow quantities:They are dependent on:(2) The Fundamental Dimensions:The fundamental dimensionsReference quantities:Specific-heat ratio of eight common gases as a function of temperatureReference quantities:Reference quantities:incompressible flow, where density effects are negligible.subsonic flow, where density effects are important but no shockwaves appear.transonic flow, where shock waves first appear, dividing subsonicand supersonic regions of the flow. Powered flight in the transonic region is difficult because of the mixed character of the flow field.supersonic flow, where shock waves are present but there are nosubsonic regions.hypersonic flow, where shock waves and other flow changes areespecially strong.incompressible flow, where density effects are negligible. subsonic flow, where density effects are important but no shockwaves appear.transonic flow, where shock waves first appear, dividing subsonicand supersonic regions of the flow. Powered flight in the transonic region is difficult because of the mixed character of the flow field.supersonic flow, where shock waves are present but there are nosubsonic regions.hypersonic flow, where shock waves and other flow changes areespecially strong.•Similarity of molecular transport processes•Similarity of flow processes•Similarity of heat transfer processesNatural convection•Similarity of integral quantities of forces•Similarity of integral quantities of heat and mass transferheat transfer coefficientmass transfer coefficientReynolds number(1) Flow constructureReynolds number, as the similarity parameter of viscous flow, has very important position in analyzing the flow.Reynolds number (2) Reynolds number and Drag(2) Reynolds number and Drag(2) Reynolds number and Drag(2) Reynolds number and DragTime history of Aerodynamic drag of cars in comparison withchange in geometry of streamlined bodies (bluff to streamline).Reynolds number(3) Reynolds number and the flight mode (Fixed wing?)。