# AA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow

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1 AA210A Fundamentals of Compressible Flow Chapter 1 - Introduction to fluid flow 1

2 1.2 Conservation of mass Mass flux in the x-direction [ ρu ] = M L 3 L T = M L 2 T Momentum per unit volume Mass per unit area per second 2

3 Divide through by the volume of the control volume Conservation of mass - Incompressible flow If the density is constant the continuity equation reduces to Note that this equation applies to both steady and unsteady incompressible flow 3

4 1.2.2 Index notation and the Einstein convention Make the following replacements Using index notation the continuity equation is Einstein recognized that such sums from vector calculus always involve a repeated index. For convenience he dropped the summation symbol. Coordinate independent form 4

5 1.3 Particle paths, streamlines and streaklines in 2-D steady flow The figure below shows the streamlines over a 2-D airfoil. The flow is irrotational and incompressible 5

6 Streamlines Streaklines 6

7 A vector field that satisfies can always be represented as the gradient of a scalar potential or If the scalar potential is substituted into the continuity equation the result is Laplaces equation. 7

8 A weakly compressible example - flow over a wing flap. 8

9 Particle paths The figure below shows the trajectory in space of a fluid element moving under the action of a two-dimensional steady velocity field The equations that determine the trajectory are: 9

10 Formally, these equations are solved by integrating the velocity field in time. Along a particle path 10

11 Eliminate time between the functions F and G to produce a family of lines. These are the streamlines observed in the figures shown earlier. The value of a particular streamline is determined by the initial conditions. 11

12 This situation is depicted schematically below. 12

13 The streamfunction can also be determined by solving the first-order ODE generated by eliminating dt from the particle path equations. The total differential of the streamfunction is 13

14 Replace the differentials dx and dy. The stream function, can be determined as the solution of a linear, first order PDE. This equation is the mathematical expression of the statement that streamlines are parallel to the velocity vector field. 14

15 The first-order ODE governing the stream function can be written as The integrating factor On a streamline What is the relationship between these two equations? 15

16 To be a perfect differential the functions U and V have to satisfy the integrability condition For general functions U and V this condition is not satisfied. The equation must be multiplied by an integrating factor in order to convert it to a perfect differential. It was shown by the German mathematician Johann Pfaff in the early 1800 s that an integrating factor M(x,y) always exists. and the partial derivatives are 16

17 1.3.2 Incompressible flow in 2 dimensions The flow of an incompressible fluid in 2-D is constrained by the continuity equation This is exactly the integrability condition. Continuity is satisfied identically by the introduction of the stream function, In this case -Vdx+Udy is guaranteed to be a perfect differential and one can write Incompressible, irrotational flow in 2 dimensions The Cauchy-Reimann conditions 17

18 1.3.4 Compressible flow in 2 dimensions The continuity equation for the steady flow of a compressible fluid in two dimensions is In this case the required integrating factor is the density and we can write. The stream function in a compressible flow is proportional to the mass flux and the convergence and divergence of lines in the flow over the flap shown earlier is a reflection of variations of mass flux over different parts of the flow field. 18

19 1.4 Particle paths in three dimensions The figure above shows the trajectory in space traced out by a particle under the action of a general three dimensional unsteady flow, 19

20 The equations governing the motion of the particle are: Formally, these equations are solved by integrating the velocity field. 20

21 1.5 The substantial derivative The acceleration of a particle is Insert the velocities. The result is called the substantial or material derivative and is usually denoted by The time derivative of any flow variable evaluated on a fluid element is given by a similar formula. For example the rate of change of density following a fluid particle is 21

22 1.5.1 Frames of reference Transformation of position and velocity Please correct printed notes Transformation of momentum 22

23 Transformation of kinetic energy Thermodynamic properties such as density, temperature and pressure do not depend on the frame of reference. 23

24 1.6 Momentum transport due to convection Density Volume flux in the y direction Outward unit normal vector Control volume surface [ ρ] = M L 3 U [ V ] = L T = L3 L 2 T = Volume Area Sec V y x U Momentum flux [ ρuv ] = M L 3 L T L T = L T M L 2 T x-momentum per unit volume Volume per unit area per second x-momentum convected in the y-direction per unit area per second 24

25 The conservation equation for momentum 25

26 Divide through by the volume x - component In the y and z directions 26

27 In index notation the momentum conservation equation is Rearrange Please correct printed notes In words, 27

28 1.7 Momentum transport due to molecular motion Pressure Viscous friction - Plane Couette Flow Force/Area needed to maintain the motion of the upper plate 28

29 1.7.3 A question of signs Newtonian fluids Forces acting on a fluid element 29

30 Pressure-viscous-stress force components Momentum balance in the x-direction 30

31 Divide by the volume x - component In the y and z directions 31

32 In index notation the equation for conservation of momentum is Coordinate independent form 32

33 1.7 Conservation of energy 33

34 1.8.1 Pressure and viscous work Fully written out this relation is 34

35 The previous equation can be rearranged to read in terms of energy fluxes. 35

36 Energy balance. 36

37 In index notation the equation for conservation of energy is Coordinate independent form 37

38 1.9 Summary - the equations of motion 38

39 Some remarks on the pressure field Two dimensional steady, inviscid, incompressible flow Conservation of mass U x + V y = 0 Conservation of momentum U U x + V U y = x U V x + V V y = y P ρ P ρ Vorticity Ω = V x U y 39

40 x y x y For any steady, inviscid, incompressible, irrotational velocity field the pressure field exists! P ρ = U U x + V U y P ρ = U V x + V V y P ρ U 2 + V 2 ( ) P ρ U 2 + V 2 ( ) = 0 = 0 P ρ U 2 + V 2 ( ) = const Ω = V x U y = 0 = U U x V V x = 1 2 = U U y V V y = 1 2 x U 2 + V 2 ( ) y U 2 + V 2 ( ) This is the incompressible Bernoulli pressure 40

41 1.10 Problems 41

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