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Incompressible Flow AA210A Fundamentals of Compressible Flow Chapter 1 - Introduction to fluid flow 9/15/20 1 1.2 Conservation of mass Mass flux in the x-direction L M ⎛ ⎞ ⎝⎜ T ⎠⎟ M ρU = = [ ] L3 L2T Momentum Mass per per unit unit area volume per second 9/15/20 2 Divide through by the volume of the control volume. 1.2.1 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 9/15/20 3 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 9/15/20 4 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 9/15/20 5 Streamlines Streaklines 9/15/20 6 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. 9/15/20 7 A weakly compressible example - flow over a wing flap. 9/15/20 8 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: 9/15/20 9 Formally, these equations are solved by integrating the velocity field in time. Along a particle path 9/15/20 10 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. 9/15/20 11 This situation is depicted schematically below. 9/15/20 12 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 9/15/20 13 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. 9/15/20 14 The first-order ODE governing the stream function can be written as 1.3.1 The integrating factor On a streamline What is the relationship between these two equations ? 9/15/20 15 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 9/15/20 16 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. 1.3.3 Incompressible, irrotational flow in 2 dimensions The Cauchy-Reimann conditions 9/15/20 17 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. 9/15/20 18 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, 9/15/20 19 The equations governing the motion of the particle are: Formally, these equations are solved by integrating the velocity field. 9/15/20 20 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 9/15/20 21 1.5.1 Frames of reference Transformation of position and velocity Transformation of momentum 9/15/20 22 Transformation of kinetic energy Thermodynamic properties such as density, temperature and pressure do not depend on the frame of reference. 9/15/20 23 1.6 Momentum transport due to convection Density Volume flux in the y direction M L L3 Volume [ρ] = 3 [V ] = = 2 = L T L T Area ⋅Sec Outward unit normal vector y U V Control volume surface x U Momentum flux ⎛ L ⎞ ⎛ L ⎞ M ⎜ ⎟ M ⎜ ⎟ ⎝ T ⎠ ⎛ L ⎞ ⎝ T ⎠ ρUV = = [ ] L3 ⎝⎜ T ⎠⎟ L2T Volume x-momentum x-momentum convected in the per unit per unit area per y-direction per unit area per volume second second 9/15/20 24 The conservation equation for momentum 9/15/20 25 Divide through by the volume x - component In the y and z directions 9/15/20 26 In index notation the momentum conservation equation is Rearrange In words, 9/15/20 27 1.7 Momentum transport due to molecular motion 1.7.1 Pressure 1.7.2 Viscous friction - Plane Couette Flow Force/Area needed to maintain the motion of the upper plate 9/15/20 28 1.7.3 A question of signs 1.7.4 Newtonian fluids 1.7.5 Forces acting on a fluid element 9/15/20 29 Pressure-viscous-stress force components Momentum balance in the x-direction 9/15/20 30 Divide by the volume x - component In the y and z directions 9/15/20 31 In index notation the equation for conservation of momentum is Coordinate independent form 9/15/20 32 1.7 Conservation of energy 9/15/20 33 1.8.1 Pressure and viscous work Fully written out this relation is 9/15/20 34 The previous equation can be rearranged to read in terms of energy fluxes. 9/15/20 35 Energy balance. 9/15/20 36 In index notation the equation for conservation of energy is Coordinate independent form 9/15/20 37 1.9 Summary - the equations of motion 9/15/20 38 Some remarks on the pressure field Two dimensional steady, inviscid, incompressible flow Conservation of mass ∂U ∂V + = 0 ∂x ∂y Conservation of momentum ∂U ∂U ∂ ⎛ P⎞ U +V = − ∂x ∂y ∂x ⎝⎜ ρ ⎠⎟ ∂V ∂V ∂ ⎛ P⎞ U +V = − ∂x ∂y ∂y ⎝⎜ ρ ⎠⎟ Vorticity ∂V ∂U Ω = − ∂x ∂y 9/15/20 39 For any steady, inviscid, incompressible, irrotational velocity field the pressure field exists! ∂V ∂U Ω = − = 0 ∂x ∂y ∂ ⎛ P⎞ ⎛ ∂U ∂U ⎞ ∂U ∂V 1 ∂ = − U +V = −U −V = − U 2 +V 2 ∂x ⎝⎜ ρ ⎠⎟ ⎝⎜ ∂x ∂y ⎠⎟ ∂x ∂x 2 ∂x ( ) ∂ ⎛ P⎞ ⎛ ∂V ∂V ⎞ ∂U ∂V 1 ∂ = − U +V = −U −V = − U 2 +V 2 ∂y ⎝⎜ ρ ⎠⎟ ⎝⎜ ∂x ∂y ⎠⎟ ∂y ∂y 2 ∂y ( ) ∂ ⎛ P 1 ⎞ + U 2 +V 2 = 0 ∂x ⎝⎜ ρ 2 ( )⎠⎟ ∂ ⎛ P 1 ⎞ + U 2 +V 2 = 0 ∂y ⎝⎜ ρ 2 ( )⎠⎟ P 1 + U 2 +V 2 = const ρ 2 ( ) This is the incompressible Bernoulli pressure 9/15/20 40 1.10 Problems 9/15/20 41 9/15/20 42 9/15/20 43 9/15/20 44 9/15/20 45.
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