《空气动力学》（双语版）Chapter 3 Fundamentals of Inviscid, incompressible Flow

Substituting the differential equations of the streamline into x component momentum equation Ov +u=dz u-dx+u-dv az p Ox or ou dx+dv+ 1 op dx p Or as dx+dy+dz Ox az

Substituting the differential equations of the streamline into x component momentum equation dx x p dz z u dy u y u dx u x u u   = −   +   +    1 or dx x p dz z u dy y u dx x u u   = −           +   +    1 as dz z u dy y u dx x u du   +   +   =

We have p ox O厂 i dual op dx 2 O In the same way we can get z 2 2 P Oz

We have dx x p udu   = −  1 or dx x p du   = −  1 2 1 2 In the same way, we can get dy y p dv   = −  1 2 1 2 dz z p dw   = −  1 2 1 2

d(12+y2+)s1(cax dy+dz 2 0(O as L12+12+u2=r2 ana dx+ dy dz= dp OX az Then we have dv2- 1 dp or dp=-pvdy 2

          +   +   + + = − dz z p dy y p dx x p d u v w  1 ( ) 2 1 2 2 2 2 2 2 2 u + v + w =V as and dz dp z p dy y p dx x p =   +   +   Then, we have dV dp  1 2 1 2 = − or dp = −VdV

dp=-prdv The above equation is called Fuler's equation precondition: inviscid, without body force, along a streamline usage: setup the relation between dv and dp Step 2, integration of Euler's equation For incompressible flows, p=const The integration from point l to point 2 along a streamline is

dp = −VdV The above equation is called Euler’s Equation. precondition: inviscid, without body force, along a streamline. usage: setup the relation between dV and dp Step 2. integration of Euler’s equation For incompressible flows, The integration from point 1 to point 2 along a streamline is  = const   = − 2 1 2 1 V V p p dp  VdV

2 2 O厂 P2-P1==P 22 O厂 n2+,2=n1+PV2 The above equation is called Bernoullis equation precondition: steady, inviscid, incompressible, without body force, along a streamline usage: setup the relation between Vi and p, at point 1 on a streamline to v, and p, at another point 2 on the same streamline

or         − = − − 2 2 2 1 2 2 2 1 V V p p  or 2 1 1 2 2 2 2 1 2 1 p + V = p + V The above equation is called Bernoulli’s Equation. precondition: steady, inviscid, incompressible, without body force, along a streamline. usage: setup the relation between at point 1 on a streamline to at another point 2 on the same streamline. V1 and p1 V2 and p2