空气动力学电子教案Chapter.ppt

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,PARTI,FUNDAMENTALPRINCIPLES,（基本原理）,InpartI,wecoversomeofthebasicprinciplesthatapplytoaerodynamicsingeneral.Thesearethepillarsonwhichallofaerodynamicsisbased,Chapter 2,Aerodynamics:Some Fundamental,Principles and Equations,There is so great a difference between a fluid and a collection of solid particles that the laws of pressure and of equilibrium of fluids are very different from the laws of the pressure and equilibrium of solids.,Jean Le Rond dAlembert,1768,2.1,Introduction and Road Map,Preparation of tools for the analysis of aerodynamics,Every aerodynamic tool we developed in this and subsequent chapters is important for the analysis and understanding of practical problems,Orientation offered by the road map,2.2,Review of Vector relations,2.2.1 to 2.2.10,Skipped over,2.2.11 Relations between line,surface,and volume integrals,The line integral of,A,over C is related to the surface integral of,A,(curl of A)over S by,Stokes theorem,:,Where aera S is bounded by the closed curve C:,The surface integral of,A,over S is related to the volume integral of,A,(divergence of A)over V by,divergence theorem,:,Where volume V is bounded by the closed surface S:,If p represents a scalar field,a vector relationship analogous to,divergence theorem,is given by gradient,theorem,:,2.3,Models of the fluid:control volumes and fluid particles,Importance to create physical feeling from physical observation.,How to make reasonable judgments on difficult problems.,In this chapter,basic equations of aerodynamics will be derived.,Philosophical procedure involved with the development of these equations,Invoke three fundamental physical principles which are deeply entrenched in our macroscopic observations of nature,namely,a.Mass is conserved,thats to say,mass can be neither created nor destroyed.,b.Newtons second law:force=mass,acceleration,c.Energy is conserved;it can only change from one form to another,2.Determine a suitable model of the fluid.,3.Apply the fundamental physical principles listed in item 1 to the model of the fluid determined in item2 in order to obtain mathematical equations which properly describe the physics of the flow.,Emphasis of this section:,What is a suitable model of the fluid?,How do we visualize this squishy substance in order to apply the three fundamental principles?,Three different models mostly used to deal with aerodynamics.,finite control volume（,有限控制体）,infinitesimal fluid element（,无限小流体微团）,molecular（,自由分子）,2.3.1,Finite control volume approach,Definition of finite control volume:,a closed volume sculptured within a finite region of the flow.The volume is called,control volume V,and the curved surface which envelops this region is defined as,control surface S,.,Fixed control volume and moving control volume.,Focus of our investigation for fluid flow.,2.3.2,Infinitesimal fluid element approach,Definition of infinitesimal fluid element:,an infinitesimally small fluid element in the flow,with a differential volume.,It contains huge large amount of molecules,Fixed and moving infinitesimal fluid element.,Focus of our investigation for fluid flow.,The fluid element may be fixed in space with fluid moving through it,or it may be moving along a streamline with velocity V equal to the flow velocity at each point as well.,2.3.3,Molecule approach,Definition of molecule approach:,The fluid properties are defined with the use of suitable statistical averaging in the microscope wherein the fundamental laws of nature are applied directly to atoms and molecules.,In summary,although many variations on the theme can be found in different texts for the derivation of the general equations of the fluid flow,the flow model can be usually be categorized under one of the approach described above.,2.3.4,Physical meaning of the divergence of velocity,Definition of :,is physically the time rate of change of the volume of a moving fluid element of fixed mass per unit volume of that element.,Analysis of the above definition:,Step 1.,Select a suitable model to give a frame under which the flow field is being described.,a moving control volume is selected,.,Step 2.,Select a suitable model to give a frame under which the flow field is being described.,a moving control volume is selected,.,Step 3.,How about the characteristics for this moving control volume?,volume,control surface and density will be changing as it moves to different region of the flow.,Step 4.,Chang in volume due to the movement of an infinitesimal element of the surface,dS,over,.,The total change in volume of the whole control volume over the time increment is obviously given as bellow,Step 5.,If the integral above is divided by,.the result is physically the time rate change of the control volume,Step 6.,Applying Gauss theorem,we have,Step 7.,As the moving control volume approaches to a infinitesimal volume,.Then the above equation can be rewritten as,Assume that is small enough such that is the same through out .Then,the integral can be approximated as ,we have,or,Definition of :,is physically the time rate of change of the volume of a moving fluid element of fixed mass per unit volume of that element.,Another description of and :,Assume is a control surface corresponding to control volume ,which is selected in the space at time .,At time the fluid particles enclosed by at time will have moved to the region enclosed by the surface .,The volume of the group of particles with fixed identity enclosed by at time is the sum of the volume in region,A,and,B,.And at time ,this volume will be the sum of the volume in region,B,and,C,.,As time interval approaches to zero,coincides with .If is considered as a fixed control volume,then,the region in,A,can be imagined as the volume enter into the control surface,C,leave out.,Based on the argument above,the integral of can be expressed as volume flux through fixed control surface.Further,can be expressed as the rate at which fluid volume is leaving a point per unit volume.,The average value of the velocity component on the right-hand x face is,The rate of volume flow out of the right-hand x face is,That into the left-hand x face is,The net outflow from the x faces is,per unit time,The net outflow from all the faces in x,y,z directions per unit time is,The flux of volume from a point is,2.4,Continuity equation,In this section,we will apply fundamental physical principles to the fluid model.More attention should be given for the way we are progressing in the derivation of basic flow equations.,Derivation of continuity equation,Step 1.,Selection of fluid model.A fixed finite control volume is employed as the frame for the analysis of the flow.Herein,the control surface and control volume is fixed in space.,Step 2.,Introduction of the concept of,mass flow,.Let a given area,A,is arbitrarily oriented in a flow,the figure given bellow is an edge view.If,A,is small enough,then the velocity,V,over the area is uniform across,A,.The volume across the area,A,in time interval,dt,can be given as,The mass inside the shaded volume is,The,mass flow,through is defined as the mass crossing,A,per unit second,and denoted as,or,The equation above states that,mass flow,through,A,is given by the product,Area X density X component of flow velocity,normal,to the area,mass flux,is defined as the mass flow per unit area,Step 3.,Physical principle,Mass can be neither created nor destroyed,.,Step 4.,Description of the flow field,control volume and control surface.,Directional elementary surface area on the control surface,Elementary volume inside the finite control volume,Step 5.,Apply the mass conservation law to this control volume.,Net mass flow,out,of control volume through surface,S,Time rate decrease of mass inside control volume,V,or,Step 6.,Mathematical expression of,B,The elemental mass flow across the area is,The physical meaning of positive and negative of,The,net,mass flow,out,of the whole control surface,S,Step 7.,Mathematical expression of,C,The mass contained inside the elemental volume,V,is,The mass inside the entire control volume is,The time rate of,increase,of the mass inside,V,is,The time rate of,decrease,of the mass inside,V,is,Step 8.,Final result of the derivation,Let,B=C,then we get,or,Derivation with,moving control volume,Mass at time,Mass at time,Based on mass conservation law,Consider the limits as,Then we get the mathematical description of the mass conservation law with the use of moving control volume,Why the final results derived with different fluid model are the same?,Step 9.,Notes for the,Continuity Equation,above,The continuity equation above is in integral form,it gives the physical behaviour over a finite region of space without detailed concerns for every distinct point.,This feature gives us numerous opportunities to apply the integral form of continuity equation for practical fluid dynamic or aerodynamic problems.,If we want to get the detailed performance at a given point,then,what shall we deal with the integral form above to get a proper mathematic description for mass conservation law?,Step 10.,Continuity Equation,in,Differential,form,Control volume is fixed in space,The integral limit is not the same,The integral limit is the same,or,A possible case for the integral over the control volume,If the finite control volume is arbitrarily chosen in the space,the only way to make the equation being satisfied is that,the integrand of the equation must be zero at all points within the control volume.That is,That is the continuity equation in a,partial differential form,.It concerns the flow field variables at,a point,in the flow with respect to the mass conservation law,It is important to keep in mind that the continuity equations in integral form and differential form are equally valid statements of the physical principles of conservation of mass.they are mathematical representations,but always remember that they speak words.,Step 11.,Limitations of the equations derived,Continuum flow or molecular flow,As the nature of the fluid is assumed as Continuum flow in the derivation so,It satisfies only for,Continuum flow,Steady flow or unsteady flow,It satisfies both,steady,and,unsteady,flows,viscous flow or inviscid flow,It satisfies both,viscous,and,inviscid,flows,Compressible flow or incompressiblw flow,It satisfies both,Compressible,and,incompressiblw,flows,Difference between steady and unsteady flow,Unsteady flow:,The flow-field variables are a function of both spatial location and time,that is,Steady flow:,The flow-field variables are a function of spatial location only,that is,For steady flow:,For steady incompressible flow:,2.5,Momentum equation,Newtons second law,where,Force exerted on a body of mass,Mass of the body,Acceleration,Consider a finite moving control volume,the mass inside this control volume should be constant as the control volume moving through the flow field.So that,Newtons second law can be rewritten as,Derivation of momentum equation,Step 1.,Selection of fluid model.A fixed finite control volume is employed as the frame for the analysis of the flow.,Step 2.,Physical principle,Force=time rate change of momentum,Step 3.,Expression of the left side of the equation of Newtons second law,i.e.,the force exerted on the fluid as it flows through the control volume.,Two sources for this force:,Body forces:over every part of,V,2.Surface forces:over every elemental surface of,S,Body force on a elemental volume,Body force over the control volume,Surface forces over the control surface can be divided into two parts,one is due to the pressure distribution,and the other is due to the viscous distribution.,Pressure force acting on the elemental surface,Note,:indication of the negative sign,Complete pressure force over the entire control surface,The surface force due to the viscous effect is simply expressed by,Total force acting on the fluid inside the control volume as it is sweeping through the fixed control volume is given as the sum of all the forces we have analyzed,Step 4.,Expression of the right side of the equation of Newtons second law,i.e.,the time rate change of momentum of the fluid as it sweeps through the fixed control volume.,Moving control volume,Let be the momentum of the fluid within region,A,B,and,C,.for instance,At time ,the momentum inside is,At time ,the momentum inside is,The momentum change during the time interval,or,As the time interval approaches to zero,the region B will coincide with S in the space,and the two limits,Net momentum flow out of control volume across surface,S,Time rate change of momentum due to unsteady fluctuations of flow properties inside,V,The explanations above helps us to make a better understanding of the arguments given in the text book bellow,Net momentum flow out of control volume across surface,S,Time rate of change of momentum due to unsteady fluctuations of flow properties inside control volume,V,Step 5.,Mathematical description of,mass flow across the elemental area,dS,is,momentum flow across the elemental area,dS,is,The net flow of momentum out of the control volume through,S,is,Step 6.,Mathematical description of,The momentum in the elemental volume,dV,is,The momentum contained at any instant inside the control volume,V,is,Its time rate change due to unsteady flow fluctuation is,Be aware of the difference between,and,Step 7.,Final result of the derivation,Combine the expressions of the forces acting on the fluid and the time rate change due to term and ,respectively,according to Newtons second low,Its the momentum equation in integral form,Its a vector equation,Advantages for momentum equation in integral form,Step 8.,Momentum Equation,in,Differential,form,Try to rearrange the every integrals to share the same limit,gradient theorem,control volume is fixed in space,Then we get,Split this vector equation as three scalar equations with,Momentum equation in x direction is,divergence theorem,As the control volume is arbitrary chosen,then the integrand should be equal to zero at any point,that is,x direction,y direction,z direction,These equations can applied for unsteady,3D flow of any fluid,compressible or incompressible,viscous or inviscid.,Steady and inviscid flow without body forces,Eulers Equations and Navier-Stokes equations,Whether the,viscous effects are being considered or not,Eulers Equations:,inviscid flow,Navier-Stokes equations:,viscous flow,Deep understanding of different terms in continuity and momentum equations,Time rate change of mass inside control volume,Time rate change of momentum inside control volume,Net flow of mass out of the control volume through control surface,S,Net flow of volume out of the control volume through control surface,S,Net flow of momentum out of the control volume through control surface,S,Body force through out the control volume,V,Surface force over the control surface,S,What we can foresee the applications for aerodynamic problems with basic flow equations on hand?,If the,stea