Why does the author not add the pressure of the freestream to the variables (dimensional analysis)?

This page (page 35) is from "Fundamentals of aerodynamics" by John D. Anderson, Jr (Fifth edition):

Why does the author not add pressure of the freestream (static pressure for clarify) to the variables ? How does he know that pressure of the freestream not effect the aerodynamic force R?

• stop posting question that are only images. type the relevant parts here. equations can be typed with mathjax – Federico Jun 16 '17 at 10:14
• Thank you for your concern. But I think this kind of question is better when I just post the image of the text, it assure the original of the text. I mean: no need to deal with formulars and special texts with this kind of theoretical question. Please give me more comment if I am wrong about this – Dat Jun 16 '17 at 10:48
• no, it is not better and we said it multiple times. Images of text cannot be searched in the future if we ever want to reference this question to other users, making it less useful. – Federico Jun 16 '17 at 10:53
• Oh I understood: the question needs to be readable by the seach machine to help other people to find the same question, right?? I am so sorry for this, I swear I will change in next question – Dat Jun 16 '17 at 10:59

Freestream static pressure is not an independent variable when both density and speed of sound are given already. Adding pressure will not improve the equation because it can be canceled out at a later stage.

Static pressure is linked to density and temperature via the ideal gas law. Temperature can be deduced from the speed of sound.

• that's good point, I also noticed that but there still is a little confuse. That's true for gases. In the liquid, you can easily change the static pressure (adding more pressure on the surface of the liquid) and keep the density and temperature stay the same. So pressure in the liquid is independent, right ?? – Dat Jun 16 '17 at 13:40
• @DatXaLin: Liquids are less compressible than gasses, but not completely incompressible. But you are right, my answer was written with only gasses at low or moderate pressure in mind. – Peter Kämpf Jun 16 '17 at 15:02

Static pressure is everywhere. At the front of the wing where the air streams onto, but also at the back of the wing.

We're talking on an intuitive basis here, and we do realise that the denser the fluid, the higher the force. Lift will be higher in water than in air, it is 1000 x denser. But it does not increase if we go from 10 m below sea level (pressure = 2 bar) to 20 m deep (pressure = 3 bar).

Static pressure will become a factor when compressibility needs to ne considered, near transsonic speeds and above. How much the air is compressed does depend on static pressure. The analysis with density and velocity only holds for lower free stream speeds where the air can be considered incompressible. Indeed, point 5 takes this into account.

Update

Friction drag of a fluid is caused by the viscosity of the fluid. According to Viscopedia, static pressure has a very small influence on the viscosity of fluids that may be considered as incompressible. Air at lower Mach number is usually considered incompressible.

• Here we have aerodynamic force R is lift and drag. Maybe the lift is not increased when you go under below, but due to pressure increasing, the drag may be different because the fluid molecules press harder to the surface of the body. And not like you say in second paragraph,the compressiblility is already considered by the 5th variable: the freestream speed of sound, not the static pressure – Dat Jun 16 '17 at 4:41
• molecules pressing harder on the surface would be skin drag, not induced drag, and therefore not part of R – Radu094 Jun 16 '17 at 6:33
• @DatXaLin ..but due to pressure increasing,..That's dynamic pressure. Static pressure is everywhere, also on the back of the wing, pressing forward. It's a random movement of molecules, in all directions. Dynamic pressure is a velocity component superimposed on the random movement. Total pressure is the sum of dynamic pressure and static pressure and is measured on the leading edge of the wing. And yes, variable 5 takes compressibility into account so for the dimensional analysis that is enough. – Koyovis Jun 16 '17 at 7:41
• @Radu094 that's not a good idea, skin drag still be the part of aerodynamic force R – Dat Jun 16 '17 at 10:33
• @Koyovis "but due to pressure increasing", I meant that's stactic pressure, not dynamic pressure. When you go below the sea level, the static pressure increase, right ?? And like I comment before, this pressure will press harder to the surface of the body, then the drag force maybe change. That's why I think the static pressure of the fluid need to take into account. – Dat Jun 16 '17 at 10:42