Determining Terminal Velocity of Aircraft

[Stoney]

I was curious as to whether terminal velocity of an aircraft can be computed on paper, and if so, what methodology would be used to determine it.  I assume the zero-lift coefficient of drag and the weight would be the two largest characteristics that impact it?

[hitech]

I was curious as to whether terminal velocity of an aircraft can be computed on paper, and if so, what methodology would be used to determine it.  I assume the zero-lift coefficient of drag and the weight would be the two largest characteristics that impact it?

Stoney, are you having a bad hair day?

The equation is simply Weight + Thrust = Drag.

Also depending what you are looking for Thrust can be negative.

HiTech

[Stoney]

Stoney, are you having a bad hair day?

The equation is simply Weight + Thrust = Drag.

Also depending what you are looking for Thrust can be negative.

HiTech

Sorry.  Sometimes the stuff I know for one thing, I have a hard time rearranging for something else I guess...

I was thinking about power off and power on maximum dive speeds.

[Rino]

     I was just thinking that terminal velocity of an aircraft is always 0.  When you get to the ramp, or the pieces stop falling 

[Brooke]

I think it's difficult to compute the terminal velocity based on the subsonic equations because they cease to be good approximations as you start hitting shockwave problems of various sorts (compressibility, prop tips going supersonic).

[Ardy123]

I think it's difficult to compute the terminal velocity based on the subsonic equations because they cease to be good approximations as you start hitting shockwave problems of various sorts (compressibility, prop tips going supersonic).

So what your saying is that there is a discontinuity in the function that describes drag as you hit supersonic speeds? Although I agree with your statement, I'm kinda baffled on how scientists have overcome this in calculating 'drag coefficients' etc...

[Brooke]

Equations like D = 0.5 * rho * v^2 * C_D are developed for and are good approximations for subsonic flow.  It is no longer a good approximation at mach 1, and people don't calculate the C_D used in this equation to be valid for mach 1 and above.  For supersonic flow, different equations become the good approximation.

[Stoney]

So what your saying is that there is a discontinuity in the function that describes drag as you hit supersonic speeds? Although I agree with your statement, I'm kinda baffled on how scientists have overcome this in calculating 'drag coefficients' etc...

Yeah, like Brooke said, most aero theory just separates the two into sub and super sonic.  Things get pretty tricky mathematically at the transition.  And Brooke, I've never seen anything above about .87 Mach in game.  Is that close enough to the transonic range to cause a problem?

[Brooke]

Yeah, like Brooke said, most aero theory just separates the two into sub and super sonic.  Things get pretty tricky mathematically at the transition.  And Brooke, I've never seen anything above about .87 Mach in game.  Is that close enough to the transonic range to cause a problem?

It is in that it is above the critical mach number for the aircraft.  At critical mach, flow over some part of the airframe has gone supersonic (usually the wing), and lift and drag become radically different than just below critical mach.

[drgondog]

Equations like D = 0.5 * rho * v^2 * C_D are developed for and are good approximations for subsonic flow.  It is no longer a good approximation at mach 1, and people don't calculate the C_D used in this equation to be valid for mach 1 and above.  For supersonic flow, different equations become the good approximation.

Brooke - I know you meant that while the equation holds, C_D loads up on Parasite drag and the 'simple' calculation of D=Dp + Di using "Dp=zero lift drag for the entire airframe" in subsonic, low AoA envelope gets very interesting in the drag rise region of Mcrit as shock wave form and wake drag 'jumps'.

That is why those that attempt to calculate Dp at peak speed for a/c like Mustangs and T-Bolts and Me 109K at say 25,000 feet will be getting 'zero lift drag of airframe at sub Mcrit' PLUS Dcomp... and wonder why zero lift drag of the airframe at top speed is lower on the deck when it is below Mcrit and there is no Dcomp.

[dtango]

So what your saying is that there is a discontinuity in the function that describes drag as you hit supersonic speeds? Although I agree with your statement, I'm kinda baffled on how scientists have overcome this in calculating 'drag coefficients' etc...

Understanding the assumptions behind the aerodynamics is the key to unlock this riddle.  The usual equations for lift and drag assume inviscid incompressible flow.  This is just a simplification to make the problem easier to solve for situations which apply that are many in aerodynamics.  The reality is that air is a compressible (and viscous & not inviscid) fluid which aerodynamicists have known about for a long time. 

Simply put incompressible flow reduces the math down to two dependent variables of pressure and velocity.  Compressible flow however re-introduces other thermodynamic dependent variables of density, internal energy, and temperature.  For incompressible flow these are assumed constant for a fixed altitude.  In reality they are not because air is compressible.  The traditional continuity, momentum, and energy equations from where Bernoulli and others developed the oft used lift & drag equations from contain all these variables.  However to simplify the math we make the incompressible assumptions which make the maths easier to solve because we ignore the complications due to thermodynamics.

There are host of ways all this is re-introduced when needed by aeros.

For subsonic compressible flows for instance the Prandtl-Glauert rule (known in the 1930's) provides compressibility corrections to lift and drag.

For transonic flows since the 1960's CFD was introduced to begin to accurately account for compressible flow.  Prior to that empirical wind tunnel & flight test data provided the data that was curve fitted and fared to predict drag.

For supersonic linearized solutions and numerical CFD solutions exist.

It gets even more interesting for the hypersonic case!

Hope you find that interesting!

[Ardy123]

Understanding the assumptions behind the aerodynamics is the key to unlock this riddle.  The usual equations for lift and drag assume inviscid incompressible flow.  This is just a simplification to make the problem easier to solve for situations which apply that are many in aerodynamics.  The reality is that air is a compressible (and viscous & not inviscid) fluid which aerodynamicists have known about for a long time. 

Simply put incompressible flow reduces the math down to two dependent variables of pressure and velocity.  Compressible flow however re-introduces other thermodynamic dependent variables of density, internal energy, and temperature.  For incompressible flow these are assumed constant for a fixed altitude.  In reality they are not because air is compressible.  The traditional continuity, momentum, and energy equations from where Bernoulli and others developed the oft used lift & drag equations from contain all these variables.  However to simplify the math we make the incompressible assumptions which make the maths easier to solve because we ignore the complications due to thermodynamics.

There are host of ways all this is re-introduced when needed by aeros.

For subsonic compressible flows for instance the Prandtl-Glauert rule (known in the 1930's) provides compressibility corrections to lift and drag.

For transonic flows since the 1960's CFD was introduced to begin to accurately account for compressible flow.  Prior to that empirical wind tunnel & flight test data provided the data that was curve fitted and fared to predict drag.

For supersonic linearized solutions and numerical CFD solutions exist.

It gets even more interesting for the hypersonic case!

Hope you find that interesting!

yeah, I do find it interesting. Im just curious, because some of the planes flown (ie 262 etc..) can get way up there, close to mach 1 (esp in a dive). As you stated, its not a discontinuity, its just that at slower speed air's compressibility is not taken into account for simplicity. That being said, I would almost assume there would be a gradual loss of control as the speed approached a  point where the gas compressed and created a vacuums around the control surfaces so great that they failed to work.

[dtango]

yeah, I do find it interesting. Im just curious, because some of the planes flown (ie 262 etc..) can get way up there, close to mach 1 (esp in a dive). As you stated, its not a discontinuity, its just that at slower speed air's compressibility is not taken into account for simplicity. That being said, I would almost assume there would be a gradual loss of control as the speed approached a  point where the gas compressed and created a vacuums around the control surfaces so great that they failed to work.

Several points worth mentioning:

1) The reason different WW2 planes experience compressibility effects at different airspeeds is because critical mach when shock waves form depend much on the chord shape and sweep of the wing.  For an airplane in flight air flows faster over curvatures which means it has to travel faster than the oncoming free stream of air.  Thus it's possible for airflow to reach supersonic speeds at local points on the airplane before it's actual speed of travel does.  Thinner or laminar airfoils and wing sweep essentially delay the onset of when airflow reaches local supersonic speeds.

2) As for gradual loss of control, it doesn't occur this way because nature's thermodynamic effects are nonlinear.  For instance subsonic drag rise from compressible flow is shaped more like an exponential curve where it's nearly flat for good while before rising sharply as it increases with airspeed.  

2ndly the formation of shock waves as local flow become supersonic are sudden events resulting in dramatic changes between pressure, temperature, density, and velocity of air in front and behind the shock wave.  Thus shock waves and boundary layer flows around objects don't mix well resulting in sudden boundary layer separation which results in the various types of loss of control.  The following illustration demonstrates this:



Image source: http://www.langleyflyingschool.com/Pages/CPGS%204%20Aerodynamics%20and%20Theory%20of%20Flight%20Part%201.html