Density.pdf

Dynamic Pressure: An important property of any gas is its pressure. Because understanding what pressure is and how it works is so fundamental to the understanding of aerodynamics and propulsion. There are two ways to look at pressure: (1) the small scale action of individual air molecules or (2) the large scale action of a large number of molecules. On the the small scale, from the kinetic theory of gases, a gas is composed of a large number of molecules that are very small relative to the distance between molecules. The molecules of a gas are in constant, random motion and frequently collide with each other and with the walls of any container. During collisions with the walls, there is a change in velocity and therefore a change in momentum of the molecules. The change in momentum produces a force on the walls which is related to the gas pressure. The pressure of a gas is a measure of the average linear momentum of the moving molecules of a gas. On the large scale, the pressure of a gas is a state variable, like the temperature and the density. The change in pressure during any process is governed by the laws of thermodynamics. Although pressure itself is a scalar quantity, we can define a pressure force to be equal to the pressure (force/area) times the surface area in a direction perpendicular to the surface. If a gas is static and not flowing, the measured pressure is the same in all directions. But if the gas is moving, the measured pressure depends on the direction of motion. This leads to the definition of the dynamic pressure.

To understand dynamic pressure, let’s begin with a one dimensional version of the conservation of linear momentum for a fluid. r * u * du/dx = - dp/dx Where r is the density of the gas, p is the pressure, x is the direction of the flow, and u is the velocity in the x direction. Performing a little algebra:

dp/dx + r * u * du/dx = 0 For a constant density (incompressible flow) take the "r * u" term inside the differential: dp/dx + d(.5 * r * u^2)/dx = 0 And then gather all of the terms: d(p + .5 * r * u^2)/dx = 0 Integrating this differential equation: ps + .5 * r * u^2 = constant = pt This equation looks exactly like the incompressible form of Bernoulli's equation. Each term in this equation has the dimensions of a pressure (force/area); ps is the static pressure, the constant pt is called the total pressure, and

.5 * r * u^2 Is called the dynamic pressure because it is a pressure term associated with the velocity u of the flow. Dynamic pressure is often assigned the letter q in aerodynamics: q = .5 * r * u^2 The dynamic pressure is a defined property of a moving flow of gas. The performance of this simple derivation determines the form of the dynamic pressure, but it can be used and applied the idea of dynamic pressure in much more complex flows, like compressible flows or viscous flows. In particular, the aerodynamic forces acting on an object as it moves through the air are directly proportional to the dynamic pressure. The dynamic pressure is therefore used in the definition of the lift coefficient and the drag coefficient. As seen, dynamic pressure appears in Bernoulli's equation even though that relationship was originally derived using energy conservation. By measuring the dynamic pressure in flight, a pitot-static tube (Prandtl tube) can be used to determine the airspeed of an aircraft. EXERCISE: Climbing flight: The drag polar of a turbojet airplane is given by: C_D=C_D0+K C_L^2. Assuming the thrust is independent of flight speed, show that the dynamic pressure when the rate of climb is maximum is given by

Hint: Calculate the dimensionless speed for the maximum rate of climb and apply definition of dynamic pressure. SOLUTION: Analytical solution for climbing flight of propeller aircraft. For propeller aircraft whose power developed by the engine P(kW) and propulsive efficiency np are independent of flight velocity. Eq.(2.104) takes form.

For maximum R/C,

It may be observed that the speed at wich the rate of climb is maximum is the same as the speed at wich the power required in level flight is minimum. The two speeds are the same because, for the propeller aircraft, we have assumed that the power available is independent of forward speed. Thus, essentially, the speed at which excess power is maximum becomes the same as that when the power required is minimum,i.e, Vr/c,max=Vmp

The climb angle is given by:

For steepest climb, i.e

This equation is of the form V^4 + aV+b=0, where

The equation has no closed form analytical solution. We have to obtain either a numerical or a graphical solution to determine ymax. For this purpose, let us plot R/C against velocity as shown in fig 2.15. This plot is called the hodograph of climbing flight. A hodograph is a graph in which the variation of one velocity component is plotted against the other. The maximum climb angle occurs at that velocity when the excess thrust per unit weight is maximum. This corresponds to the point where a line drawn from the origin is tangent to the hodograph as shown in fig 2.15. The time to climb from a given initial altitude h1 to final altitude hf is given by

Which is equal to the area under the curve obtained by plotting inverse specific excess power against altitude as shown in fig. 2.16. A special case of interest in climbing flight is the minimum time to climb.

REFERENCES: https://www.grc.nasa.gov/www/k-12/airplane/dynpress.html Perfonmance-Stability-Dynamics-and-Control-Airplane-Bandu-n-Pamadi.pdf