PART II INVISCID INCOMPRESSIBLE FLOW In part Il, we deal with the flow of a fluid which has constant density- incompressible flow. This applies to the flow of liquids, such as water flow, and to low-speed flow of gases. The material covered here is applicable to low peed flight through the atmosphere--flight at a Mach number of about 0.3
PART II INVISCID INCOMPRESSIBLE FLOW In part II, we deal with the flow of a fluid which has constant density– incompressible flow. This applies to the flow of liquids, such as water flow, and to low-speed flow of gases. The material covered here is applicable to lowspeed flight through the atmosphere—flight at a Mach number of about 0.3
Chapter 3 Fundamentals of inviscid incompressible Flow Theoretical fluid dynamics, being a difficult subject, is for convenience, commonly divided into two branches, one treating frictionless or perfect fluids, the other treating of viscous or imperfect fluids. The frictionless fluid has no existence in nature, but is hypothesized by mathematicians in order to facilitate the investigation of important laws and principles that may be approximately true of viscous or natural fluids
Chapter 3 Fundamentals of Inviscid, incompressible Flow Theoretical fluid dynamics, being a difficult subject, is for convenience, commonly divided into two branches, one treating frictionless or perfect fluids, the other treating of viscous or imperfect fluids. The frictionless fluid has no existence in nature, but is hypothesized by mathematicians in order to facilitate the investigation of important laws and principles that may be approximately true of viscous or natural fluids
3.1 Introduction and Road Map o From an aerodynamic point of view, at air velocities between 0 to 360km/h the air density remains essentially constant varying only a few percent o Purpose of this chapter is to establish some fundamental relations applicable to inviscid, incompressible flows and to discuss some simple but important flow fields and applications
3.1 Introduction and Road Map From an aerodynamic point of view, at air velocities between 0 to 360km/h the air density remains essentially constant, varying only a few percent. Purpose of this chapter is to establish some fundamental relations applicable to inviscid, incompressible flows and to discuss some simple but important flow fields and applications
3.2 Bernoulli's Equation p+pv=const o Derivation of Bernoulli,s equation Step 1. X component momentum equation without viscous effect and body forces D Dt O厂 Ou op -+m-+m-+ at z ax
3.2 Bernoulli’s Equation p + V = const 2 2 1 Derivation of Bernoulli’s equation Step 1. x component momentum equation without viscous effect and body forces x p Dt Du = − or x p z u w y u v x u u t u = − + + +
For steady flow, Ou/at=0, then +1—+v ax ay az p Ox Multiply both sides with dx ou ou pu-dx+pv dx+ow- dx Along a streamline in 3D space there are udz-wdx=0 vdx-udy=0
For steady flow, , then u t = 0 x p z u w y u v x u u = − + + 1 Multiply both sides with dx dx x p dx z u dx w y u dx v x u u = − + + Along a streamline in 3D space, there are 0 0 − = − = vdx udy udz wdx
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