空气动力学基础英文版第六章

1、Chapter 6Three-Dimensional pressible Flow6.1 IntroductionThe purpose of this chapter is to introduce some very basic considerations of three-dimensional pressible flow. Its function is simply to open the door to the analysis of three-dimensional flow.If the flow is irrotational, there exists a veloc

2、ity potential such thatA physically possible flow must satisfy the continuity equation, i.e., for pressible flow This leads to the Laplaces equation In spherical coordinates, the Laplaces eq. isThe solution of the above eq. must satisfy flow tangency boundary condition on the body, i.e.,where6.2 Thr

3、ee-Dimensional SourceConsider the velocity potential given bywhere C is a constant and r is the radial coordinate from the origin. This satisfies Laplaces eq., hence it describes a physically possible flow. According to the definition of the gradient in spherical coordinateswe obtainOn the other han

4、dThenThis velocity distribution describes a flow with straight streamlines emanating emneiti from the origin. Moreover, the velocity varies inversely with the square of the distance from the origin. Such a flow is defined as a three-dimensional source.Mass flow:Volume flow: (denoted by )FurtherHence

5、Finally we findIn the above equations, is defined as the strength of the source. When is negative, we have a point sink.6.3 Three-Dimensional DoubletSource-sink pair degenerates into a doubletPlace a sink at the origin (point O); The distance between the source and sink is OA=l.The sink and source a

6、re of equal but opposite strength; Place a source at A on the z axis;The potential at arbitrary point P located a distance r from the sink (O) and a distance from the source (A):orLet the source (A) approach the sink (O), i.e., l0whereas with l=const =In the limit,The potential es orwhere . The flow

7、 field produced by the above eq. is a three-dimensional doublet; is defined as the strength of the doublet.The two-dimensional counterpartVelocity components:FromWe can write velocity components separatelyThere is no flow in direction , i.e., the flow is tangent to the =const plane (zr plane), in ot

8、her words, the flow is inside the zr ( =const) plane. Moreover the flows (streamlines) in different =const planes are identical. In a perspective view, the flow is characterized by a series of stream surfaces which are generated byThus the flow is symmetric about z axis axisymmetric flow.revolving t

9、he streamlines in any =const plane around the z axis.6.4 Flow over a SphereUniform flow + a doubletLet the uniform flow be in the negative z direction, i.e., opposite to the direction of the doublet.Velocity components:Uniform flow:Doublet:Then the new flow=uniform flow + doublet flow: Find the stag

10、nation points:SetStagnation points found:LetThen r=R represent the surface of a sphere.Examine the sphere r=R :That is, at r=R for all values of and . This indicates that r=R is a closed stream surface of the flow and also a dividing surface since the flow outside this surface can not feel the diffe

11、rence if it is replaced by a solid body. Thus the external flow field outside r=R can be regarded as the flow around a solid sphere of radius R pressible flow over a sphere of radius R.Usually we are more comfortable or accustomed to visualizing a freestream which moves horizontally, say, from lift

12、to right. So let us flip(翻转) the coordinate system on its side. Turn over the freestream velocity to horizontal:(Push the z axis down!)Then the final synthesized flow is in horizontal direction (negative z axis).Velocity distribution on the sphere:That is, on the sphere surfaceThe maximum velocity o

13、ccurs at the top and bottom points of the sphere:For comparison, on a circular cylinder the velocity is and the maximum velocity isFor the same , the maximum velocity on a sphere is less than that for a cylinder. The flow over a sphere is somewhat “relieved”(释放，放泄分流，泄流) in comparison with the flow o

14、ver a cylinder. The flow over a sphere has extra dimension in which to move out of the way (绕开) of the solid body; the flow can move sideways as well as up and down. In contrast, the flow over a cylinder is more constrained; it can only move up and down. Hence the maximum velocity on a sphere is les

15、s than that on a cylinder. This is an example of the three-dimensional relieving effect (三维分流泄流效应), which is a general phenomenon for all types of three-dimensional flows.Pressure distribution on the sphere:orwhereas on a circular cylinderThe absolute magnitude of the pressure coefficient on a spher

16、e is less than that for a cylinderagain, an example of the three-dimensional relieving effect.6.5 General Three-Dimensional Flows: Panel TechniquesThe general idea behind all such panel programs is to cover the three-dimensional body with panels over which there is an unknown distribution of singula

17、rities (such as point sources, doublets, or vortices). Further, the requirement that the body surface be a stream surface (flow-tangency condition) will lead to a system of simultaneous linear algebraic equations.For a non-lifting body, a distribution of source panels is sufficient.Non-lifting bodyF

18、or a lifting body, both source and vortex panels (or their equivalent) are necessary. The boundary layer over the front face of a sphere is laminar at lower Reynolds numbers, and turbulent at higher Reynolds numbers. 6.6 Applied Aerodynamics: the Flow over a SphereWhen it is laminar ( ), separation

19、starts almost as soon as the pressure gradient es adverse (very near the shoulder), and a large wake forms. When it is turbulent ( ), separation is delayed (to a point about 20past the shoulder) and the wake is correspondingly smaller. Re=15000, laminar separation occurs on the forward surface, slig

20、htly ahead of the vertical equator of the sphere.If the boundary layer of a sphere can be made turbulent at a lower Reynolds number, then the drag should also go down at that Reynolds number. This is the case, as we can show by using a trip wire. A trip wire is simply a wire located on the front fac

21、e of the sphere and it introduces a large disturbance into the boundary layer. This disturbance causes an early transition to turbulence, resulting in a thinner wake, and a decrease in total drag. Turbulent flow forced artificially at Re=30,000, with trip wire. The flow would not transit naturally at this Re without the trip.The variation of the drag coefficient with ReSphereCylinderThe variations for a sphere are qualitatively similar to that of circular cylind