空气动力学电子教案Chapter_02

1、PART IFUNDAMENTAL PRINCIPLES（基本原理）In part I, we cover some of the basic principles that apply to aerodynamics in general. These are the pillars on which all of aerodynamics is basedChapter 2Aerodynamics: Some FundamentalPrinciples and EquationsThere is so great a difference between a fluid and a col

2、lection 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, 17682.1 Introduction and Road MapPreparation of tools for the analysis of aerodynamicsEvery aerodynamic tool we develo

3、ped in this and subsequent chapters is important for the analysis and understanding of practical problemsOrientation offered by the road map2.2 Review of Vector relations2.2.1 to 2.2.10 Skipped over2.2.11 Relations between line, surface, and volume integralsThe line integral of A over C is related t

4、o the surface integral of A(curl of A) over S by Stokes theorem:SAsAddSCWhere 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:dVdVSASAWhere volume V is bounded by the closed surface S:If p r

5、epresents a scalar field, a vector relationship analogous to divergence theorem is given by gradient theorem:dVppdVSS2.3 Models of the fluid: control volumes and fluid particlesImportance to create physical feeling from physical observation.How to make reasonable judgments on difficult problems. In

6、this chapter, basic equations of aerodynamics will be derived.Philosophical procedure involved with the development of these equationsInvoke three fundamental physical principles which are deeply entrenched in our macroscopic observations of nature, namely, a. Mass is conserved, thats to say, mass c

7、an be neither created nor destroyed. b. Newtons second law: force=massacceleration c. Energy is conserved; it can only change from one form to another2. 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

8、 in order to obtain mathematical equations which properly describe the physics of the flow. wEmphasis of this section:wWhat is a suitable model of the fluid?wHow do we visualize this squishy substance in order to apply the three fundamental principles?wThree different models mostly used to deal with

9、 aerodynamics.w finite control volume （有限控制体）w infinitesimal fluid element （无限小流体微团）w molecular （自由分子） 2.3.1 Finite control volume approachwDefinition of finite control volume:w a closed volume sculptured within a finite region of the flow. The volume is called control volume V, and the curved surfa

10、ce which envelops this region is defined as control surface S.wFixed control volume and moving control volume.wFocus of our investigation for fluid flow.2.3.2 Infinitesimal fluid element approachwDefinition of infinitesimal fluid element:w an infinitesimally small fluid element in the flow, with a d

11、ifferential volume.wIt contains huge large amount of molecules wFixed and moving infinitesimal fluid element.wFocus 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 veloc

12、ity at each point as well.2.3.3 Molecule approachwDefinition of molecule approach:w 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 variation

13、s 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 velocitywDefinition of :w is physically the time rate of ch

14、ange of the volume of a moving fluid element of fixed mass per unit volume of that element.VVwAnalysis of the above definition:wStep 1. Select a suitable model to give a frame under which the flow field is being described.w a moving control volume is selected.Step 2. Select a suitable model to give

15、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 movemen

16、t of an infinitesimal element of the surface dS over . SdtVdSntVVtThe total change in volume of the whole control volume over the time increment is obviously given as bellowtSSdtVStep 5. If the integral above is divided by .the result is physically the time rate change of the control volume tSSSdVSd

17、tVtDtDV1Step 6. Applying Gauss theorem, we have VdVVDtDVStep 7. As the moving control volume approaches to a infinitesimal volume, . Then the above equation can be rewritten as VVdVVDtVDAssume that is small enough such that is the same through out . Then, the integral can be approximated as , we hav

18、e VVVVVVVDtVDDtVDVV1orwDefinition of :w is physically the time rate of change of the volume of a moving fluid element of fixed mass per unit volume of that element.VVwAnother description of and :SSdVVAssume is a control surface corresponding to control volume , which is selected in the space at time

19、 .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.

20、 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.VSt1tSt1SSt1t1SSSBased on the argument above, the integral of can be expressed as volume flux through fix

21、ed control surface. Further, can be expressed as the rate at which fluid volume is leaving a point per unit volume.SSdVVThe average value of the velocity component on the right-hand x face is)2)(xxuuThe rate of volume flow out of the right-hand x face iszyxxuu)2)(That into the left-hand x face iszyx

22、xuu)2)(The net outflow from the x faces is zyxxu)(per unit time The net outflow from all the faces in x,y,z directions per unit time is zyxzwyvxu)()()(The flux of volume from a point is zyxzyxzwyvxuVfluxinflowfluxoutflowV)()()(lim0)()()(lim0zwyvxuVfluxinflowfluxoutflowV2.4 Continuity equationwIn thi

23、s 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.wDerivation of continuity equationStep 1. Selection of fluid model. A fixed finite control volume is employed as the fr

24、ame 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

25、is uniform across A. The volume across the area A in time interval dt can be given asAdtVVolumen)(The mass inside the shaded volume isAdtVMassn)(The mass flow through is defined as the mass crossing A per unit second, and denoted as m dtAdtVmn)(or AVmnThe equation above states that mass flow through

26、 A is given by the product Area X density X component of flow velocity normal to the areamass flux is defined as the mass flow per unit areanVAmfluxMassStep 3. Physical principle Mass can be neither created nor destroyed.Step 4. Description of the flow field, control volume and control surface.),(),

27、(tzyxVVtzyx:SdDirectional elementary surface area on the control surface:dVElementary volume inside the finite control volumeStep 5. Apply the mass conservation law to this control volume.Net mass flow out of control volume through surface STime rate decrease of mass inside control volume VorCB Step

28、 6. Mathematical expression of BThe elemental mass flow across the area is SdVSdVnThe physical meaning of positive and negative of SdVSdThe net mass flow out of the whole control surface S SSdVBStep 7. Mathematical expression of CThe mass contained inside the elemental volume V is dVThe mass inside

29、the entire control volume is VdVThe time rate of increase of the mass inside V is VdVtThe time rate of decrease of the mass inside V is CdVtVStep 8. Final result of the derivation Let B=C , then we get VSdVtSdVor0SVSdVdVtDerivation with moving control volumeMass at time )()(tMtMBAMass at time t1t)()

30、(11tMtMCBBased on mass conservation law 0)()()()(11tMtMtMtMBACB 0)()()()(11tMtMtMtMACBBConsider the limits as tt 1CCBBAAdVMdVMdVM,VBBttdVttttMtM11)()(lim1SACttSdVtttMtM11)()(lim1Then we get the mathematical description of the mass conservation law with the use of moving control volume 0SVSdVdVtWhy t

31、he 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

32、 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 conservatio

33、n law?Step 10. Continuity Equation in Differential form0SVSdVdVt0SVSdVdVtControl volume is fixed in space0dVVdVtVVdVVSdVVSThe integral limit is not the sameThe integral limit is the sameor0VdVVtA possible case for the integral over the control volumeIf the finite control volume is arbitrarily chosen

34、 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,0VtThat is the continuity equation in a partial differential form. It concerns the flow field variables at a point in the flow with re

35、spect to the mass conservation lawIt 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

36、 11. Limitations of the equations derivedContinuum flow or molecular flowAs the nature of the fluid is assumed as Continuum flow in the derivation soIt satisfies only for Continuum flowSteady flow or unsteady flowIt satisfies both steady and unsteady flowsviscous flow or inviscid flowIt satisfies bo

37、th viscous and inviscid flowsCompressible flow or incompressiblw flowIt satisfies both Compressible and incompressiblw flowswDifference between steady and unsteady flowUnsteady flow:The flow-field variables are a function of both spatial location and time, that is),(),(tzyxVVtzyxSteady flow:The flow

38、-field variables are a function of spatial location only, that is),(),(zyxVVzyxFor steady flow:0t0SVSdVdVt0SSdV0Vt0VFor steady incompressible flow:0V0 V2.5 Momentum equationwNewtons second lawamFwhere:F:m:aForce exerted on a body of massmMass of the bodyAccelerationConsider a finite moving control v

39、olume, 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 asdtVmdF)(wDerivation of momentum equationStep 1. Selection of fluid model. A fixed finite control volume is employed as the frame for the a

40、nalysis of the flow. Step 2. Physical principle Force = time rate change of momentumStep 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 V1. 2

41、. Surface forces: over every elemental surface of SBody force on a elemental volume dVfBody force over the control volume VdVfSurface 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 for

42、ce acting on the elemental surfaceSpdNote: indication of the negative sign Complete pressure force over the entire control surfaceSSpdThe surface force due to the viscous effect is simply expressed by viscousFTotal force acting on the fluid inside the control volume as it is sweeping through the fix

43、ed control volume is given as the sum of all the forces we have analyzed viscousSVFSpddVfFStep 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 volumeLet be the mom

44、entum of the fluid within region A, B, and C. for instance,CBAMMM,CCBBAAdVVMdVVMdVVM,At time , the momentum inside is)()(tMtMBAtSAt time , the momentum inside is1t1S)()(11tMtMCBThe momentum change during the time interval tt 1)()()()(11tMtMtMtMBACBor )()()()(11tMtMtMtMACBBAs the time interval approa

45、ches to zero, the region B will coincide with S in the space, and the two limitsVVBBttdVtVdVVttttMtM)()()(lim111SACttVSdVtttMtM)()()(lim111SVSdV)(Net momentum flow out of control volume across surface SVdVVtTime rate change of momentum due to unsteady fluctuations of flow properties inside VThe expl

46、anations 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 STime rate of change of momentum due to unsteady fluctuations of flow properties inside control volume VGHStep 5. Mathematical description of G

47、mass flow across the elemental area dS isSdVmomentum flow across the elemental area dS isVSdVThe net flow of momentum out of the control volume through S isSVSdVGStep 6. Mathematical description of HThe momentum in the elemental volume dV isdVVThe momentum contained at any instant inside the control

48、 volume V isVdVVIts time rate change due to unsteady flow fluctuation isVdVVtHBe aware of the difference betweenVdVVtVdVVdtdandStep 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 Newton

49、s second lowHGVSdVVtVSdVdtVmdHG)(FdtVmd)(viscousSVSVFSpddVfVSdVdVVtIts the momentum equation in integral formIts a vector equationAdvantages for momentum equation in integral formStep 8. Momentum Equation in Differential formviscousSVSVFSpddVfVSdVdVVtTry to rearrange the every integrals to share the

50、 same limitVSpdVSpdgradient theoremVVdVtVdVVt)(control volume is fixed in spaceviscousVVSVFpdVdVfVSdVdVtV)(Then we getSplit this vector equation as three scalar equations with kwj vi uVMomentum equation in x direction is viscousxVVxSVFdVxpdVfuSdVdVtu)()(dVVuSdVuuSdVVSSdivergence theorem0)()(Vviscous

51、xxdVfxpVutuFAs the control volume is arbitrary chosen, then the integrand should be equal to zero at any point, that is viscousxxfxpVutu)()(FviscousxxfxpVutu)()(FviscouszzfzpVwtw)()(FviscousyyfypVvtv)()(Fx directiony directionz directionThese equations can applied for unsteady, 3D flow of any fluid,

52、 compressible or incompressible, viscous or inviscid.viscousSVVSFSpddVfdVVtVSdVSSSpdVSdVSteady and inviscid flow without body forcesxpVuypVvzpVwEules Equations and Navie-Stokes equationsWhether the viscous effects are being considered or notEules Equations: inviscid flowNavie-Stokes equations: visco

53、us flowwDeep understanding of different terms in continuity and momentum equations0SVSdVdVtviscousSVSVFSpddVfVSdVdVVtVdVtVdVVtTime ate change of mass inside contol volumeTime ate change of momentum inside contol volumeSSdVSVSdVNet flow of mass out of the contol volume though contol suface SSSdVNet f

54、low of volume out of the contol volume though contol suface SNet flow of momentum out of the contol volume though contol suface SVdVfSSpdBody foce though out the contol volume VSuface foce ove the contol suface SwWhat we can foresee the applications for aerodynamic problems with basic flow equations

55、 on hand?0 VIf the steady incompressible inviscid flows are concernedxpVu1ypVv1zpVw1Partial differential equation for velocityPartial differential equation for velocity and pressure2.6 An application of the momentum equation: drag of a 2D bodywHow to design a 2D wind tunnel test?wHow to measure the

56、lift and drag exerted on the airfoil by the fluid?A selected control volume around an airfoilDescriptions of the control volume1. The upper and lower streamlines far above and below the body (ab and hi).2. Lines perpendicular to the flow velocity far ahead and behind the body(ai and bh)3. A cut that

57、 surrounds and wraps the surface of the body(cdefg)1. Pressure at ab and hi. 2. Pressure at ai and bh . ,velocity , 3. The pressure force over the surface abhi pp ppconstantu 1)(22yuu abhiSpd4. The surface force on def by the presence of the body, this force includes the skin friction drag, and deno

58、ted as per unit span.5. The surface forces on cd and fg cancel each other. 6. The total surface force on the entire control volume isRRabhiSpdforcesurface7. The body force is negligibleApply to momentum equation, we haveRabhiSVSpdVSdVdVVtfor steady flowabhiSSpdVSdVRNote: its a vector equation.If we

59、only concern the x component of the equation, with represents the x component of .DRabhixSSpduSdVDAs boundaries of the control volume abhi are chosen far away from the body, the pressure perturbation due to the presence of the body can be neglected, that means, the pressure there equal to the freest

60、ream pressure. If the pressure distribution over abhi is constant, then0abhixSpdSo thatSuSdVDAs ab, hi, def are streamlines, then0defhiabuSdVuSdVuSdVAs cd, fg are are adjacent to each other, thenfgcduSdVuSdVThe only contribution to momentum flow through the control surface come from the boundaries a