In this section we will first convert the conservation equations to a form which is suitable to apply at each point, so we can track changes from one point to another. This involves converting from the "integral form" over a control volume or control surface, to a "differential form" which deals with small changes from point to point.

The obvious approach is to say: "if the integral, over an arbitrary control volume, of this sum of terms is zero, then this sum of terms must also be zero in the limit as I reduce the size of the control volume down to a point". Thus we can get rid of the integral signs. Unfortunately, we find that we can't yet bring everything under the same integral sign: in each of the conservation equations, there are some integrals over control volumes, and other integrals over control surfaces, and actually we would also like to know about integrals over just a closed contour in a 2-dimensional flowfield. So the first priority is to find relations between integrals over lines, surfaces and volumes.
 
 

1. Stokes Theorem and the Divergence Theorem
 
 

We have seen, in the conservation equations, , and , which are, respectively, integrals over control surfaces and control volumes.

We'll also use , the line integral over a "closed contour", which means, add up all the things we see as we walk along this line which closes on itself like a snake managing to catch its own tail.
 
 

Stokes' Theorem
 
 

where  is the vector quantity of interest, dl is the vector along the closed contour of integration c,  is the vector normal to the area enclosed by c.Now we can convert integrals over closed contours to integrals over surfaces, and vice versa.
 
 

Divergence Theorem

The divergence theorem thus lets us convert between integrals over surfaces and volumes.
 
 

Gradient Theorem

If p is a scalar (only magnitude, no direction, like pressure or density), then
 
 

Another Vector Identity: divergence of the product of a vector and scalar
 
 

where  is a scalar and  is a vector.
 
 

Substantial Derivative

So far, we derived equations for flow in a Control Volume, which is a region of space, rather than a particular clump of fluid. Now, let us see how to describe the changes to a given element of fluid. The rate of change of any property as seen by the element is :
 
 

This is the substantial derivative (see p. 118), or,

local convective
 
 

The rate of change is for two reasons:

1) things are changing rapidly at the point (or region) through which you are moving.

2) you are moving into regions where properties are different.
 
 

In the early 1980s, there was a sleet storm in Atlanta. Atlanta's roads have many ups and downs, and in those days most cars had rear-wheel drive. Snow is so rare in Atlanta that the city did not have many sand-trucks then, and people still don't have snow-tires or chains. Instead they look forward to a "snow day" when schools are closed, and employers are usually nice about letting people go home early and take their kids sliding down slopes on improvised sleds. So most offices in downtown had radios tuned to the local stations, listening to the changing weather forecast.
The local radio and TV stations had an attentive audience. They took to their "traffic-copters" and flew out west to the Georgia-Alabama border to watch the storm rolling in. So, Atlanta office workers heard of rapidly changing weather conditions, but also head that this was far out west of Atlanta. They extrapolated and believed the weather prediction that by 6pm, it would be snowing in Atlanta. Wise and considerate supervisors told their teams to go home early, to beat the expected traffic problems in the evening. Atlanta's 100,000 commuters (that was in 1980: now it would be 500,000) pulled out of their offices, and got on the roads. The traffic jam started at 2pm, but this was not a real concern, because people expected to be home by 5 at least. Unfortunately, the rate of change in the weather was not all due to the "convective term", rolling in: the temperature in Atlanta (which is several hundred feet higher than the surrounding plain) dropped, and the sleet started falling at 2pm, and kept on falling. By 3pm, 100,000 cars were on the roads, all sliding around helplessly. Every side road was blocked, because at every hill there were cars sliding across the road, unable to get up the slope. Most people did not reach home before 9pm, often in the cars of kind fellow-sufferers, after abandoning their own cars on the highway. Those who did not spend their time listening to the radio (or had nasty supervisors) just took one look at the road outside, and came back in to huddle in the office overnight.

 Morals of the story: (1) the rate of change is both local (unsteady) and convective.

(2) Don't waste time listening to weather forecasts: look outside instead.
 
 

Imagine that you are inside a TVNEWS helicopter, flying toward a tornado. You are at point (A). If you stood still at point A, you would feel the pressure decrease rapidly [ tornado going to happen there]. This is . As you move at speed, you feel other changes as you move into the pressure field of the tornado.
 
 

_________________________________

Continuity Equation: Differential Form
 

where

is called the substantial derivative.

In terms of velocity components,

Use the Vector Identity

The Continuity equation is:

The Momentum equation:

These equations are quite general (i.e., they apply to most flows that one can imagine, and are too general to get any useful results for specific problems). Exceptions occur. Imagine an immense cloud of gas jetting out of a star: this can also be described using fluid dynamics, except that one might not be able to assume that it’s a "continuum", and the velocities might be so high that they are comparable to the speed of light, so one has to consider conversions between energy and mass. Temperatures may be so high that nuclear reactions might be occurring. Under those conditions, the above equations are not general enough.

On the other hand, we have to make some specializations if we are to apply these equations to any specific problem. The most widely used form is called the "Navier-Stokes equations". These are derived by incorporating the "Stokes hypothesis" to specialize the stress terms on the right hand side to the usual case where we worry mostly about viscous stresses. We'll see more about this when we consider the forces on the right-hand side.

 Energy Equation

The energy equation, reduced to differential form, is:

Euler equation

Let's go back the momentum equation. You'll see that in most AE courses, people spend lots of time on the momentum equation. The continuity equation is pretty simple (you'll spend lots of time on that in the form of the Laplace equation later in this course), and the energy equation is usually needed only when you have to worry about heat addition, work extraction, and high-speed flow effects. The momentum equation, in its full Navier-Stokes form, is terribly difficult to solve, so everyone spends a lot of time justifying their use of simplified forms which are adequate for their purposes. The Euler equation is one such form: extremely useful. It is obtained by neglecting the viscous stress-related terms from the right hand side. In some forms of the euler equation, one may have to retain the body force terms, for example if one is dealing with flows which have severe curvature and swirl, like in a turbomachine. Away from solid surfaces or other flow boundaries, the Euler equation is usually sufficient: the viscous stresses cause major problems only close to surfaces, in most flows of interest to AEs. If we can also neglect the body force term, the Euler equation becomes quite simple, as seen below.

if we neglect body forces & viscous stresses.

In terms of Cartesian components of the vector

Incompressible, inviscid flow, no body forces

If the range of flow velocity is low, as compared to the speed of sound, you can't change the density much by changing flow velocity. Such flows are called "incompressible flows." Note that density may vary if you have other complications like heating or several gases mixing.

If density changes can be neglected, we can re-write the Euler eqns. as shown below:

Euler equation along a streamline = Bernoulli's Equation

Euler equation, written as scalar components, is:

| Steady: 

| = 
 
 

| 

Multiply by dx:

Along a streamline, by definition, the slope of the line is the same as the inclination of the velocity, i.e., the flow velocity is parallel to the streamline. Thus, dy/dx = v/u. Or,

This in fact defines the streamline direction.

constant

constant

...along a streamline.

Pressure Coefficient

We need a convenient way of expressing pressure changes from point to point in a flow. This is accomplished using the Pressure Coefficient. As seen below, the pressure coefficient tells us the size of a given pressure change as a fraction of some reference pressure. Usually the reference is the "dynamic pressure", the difference between the total or stagnation pressure, and the static pressure. As defined below, the pressure coefficient is zero at a point where the pressure is the same as the freestream static pressure. The pressure coefficient is 1.0 at a stagnation point, where the flow is brought to a halt. In places where the pressure coefficient is negative, it usually means that the flow velocity is higher than the freestream velocity.

Note:

Substituting,

Thus we can use the pressure coefficient to calculate the local velocity, given the freestream velocity.

The variation of the pressure coefficient from leading edge to trailing edge of an airfoil is shown below, for both the upper and lower surfaces. Although it does not look that way in the figure below, this airfoil is at a small positive angle of attack, so it is producing lift, upwards. The lift is obtained by integrating the difference between the pressures on the upper and lower surfaces; i.e., the lift comes from the area between the two curves in the figure.

It is more usual in the aerospace literature to plot the pressure coefficient in such a way that the negative pressure coefficients are above the horizontal axis. This is because we are usually interested in the lift, upwards. We are happy to look for more "suction", i.e., highly negative Cp values on this plot. A more practical reason for this plotting habit is that we know that the maximum positive value of Cp is restricted to 1.0, at the stagnation points (unless there is some unsteady effect, or work addition). But the suction peak can be very high: you can have Cps of -4.0, for example.

Newton's 2nd law applied to steady flow across a control surface; with viscous terms neglected:

  Bernoulli's eqn: along a streamline:

Use Divergence Theorem ¬ 

¬ 

It can be shown that (Ref. 1)

[ Ref. 1: Eskinazi, S., "Vector Mechanics of Fluids and Magnetofluids," Academic Press, NY, 1967, p. 284. ]
 

where

gives the strength of the sources in the volume v.
 

gives strength of the vortices in the volume v.

Steady constant density flow: D = 0
(in other words, there is no dilatation is the density is constant)

or acting perpendicular to both  and

This is LIFT: the aerodynamic force perpendicular to the freestream. We see that to produce lift, we must have some rotation in the flow (vorticity), with the axis of rotation at least having a component perpendicular to the freestream velocity.
 
 

Implications

a) Lift force can be explained using vorticity, and dilatation.

b) In incompressible flow, production of lift requires vorticity.

c) If you take the closed countour in the flowfield around a solid body [ and you neglected viscosity ], you would then say: no vortices inside, no dilatation = NO FORCE? [ i.e., no lift or drag ! ] This is what bothered people for a long time, and is called D'Alembert's Paradox.
 
 

4. The answer is the following (Ludwig Prandtl is generally credited with explaining this): even though we don’t see any vorticity away from the surface, there is most definitely some vorticity in the flow very close to the surface: the Boundary Layer. In this region, viscous stresses are certainly not negligible: in fact the flow must come to a halt with respect to the surface, at the surface. So, we have to include the effect of this layer when we analyze the flowfield, if we hope to be able to calculate the forces properly.

e) Now there is a different way of looking at this. Away from the flow boundaries, there is very little effect of viscosity. So if we can somehow lump all the effect of the viscosity near the surface into some convenient mechanism for producing rotation, we can then go ahead and analyze the flow, and at least calculate the lift, if not the drag, without bothering with the messy viscous stress terms in the momentum equation. In other words, we only need put in the right value of vorticity to get force perpendicular to the flow  lift force, even if you have neglected viscosity.

This is the basis of the potential flow method.

"Neglecting viscosity" is O.K. away from flow boundaries and free_shear layers if

i.e.,

So this "High Reynolds Number approximation" is quite useful for large airplanes at small angle of attack . This again is a strange and beautiful result in aerodynamics: the simple form where you neglect all the difficult terms in the momentum equation actually gives an excellent approximation of the aerodynamics of the largest airplanes! It is actually much more difficult to calculate the aerodynamics of a butterfly than of a Boeing 777.

[But we cannot get any drag in 2-D flow from this formulation. This is still OK if most of the drag on the full 3-D wing is lift-induced drag, as we saw in the first section of the course.]
 
 

Assumed:

Continuum Target:  3 unknowns

No nuclear reaction

Physical Laws:

Mass is conserved  Continuity Eqn.....(1)

Rate of change of momentum = Net Force ..Momentum Eqn....(2) 3 Equations

Energy is conserved ... dE = Dq+dW ..Energy Eqn...........(3)
 
 

Applied to entire control volumes = Integral Forms of Conservation Eqns.
 

Use Stokes; Divergence & Gradient theorems, other vector identities and substantial derivative to swap  and  and gather terms

= 

= Differential form of conservation eqns.

(1)

(2)

(3)

Momentum Eqn.

1. With a relation for shear stress as f(rate of strain), and Stokes' hypothesis (viscosity has no effect on normal stress), the momentum eqn. is called the Navier-Stokes eqn.

2. With viscous terms dropped, the momentum eqn. is called the Euler equation

Bernoulli Eqn. for Steady Incompressible Flow Along a Streamline
Integrate Euler eqn. along a streamline , steady, incompressible

= incompressible  constant
 

Kutta-Joukowski Theorem

Momentum eqn. = steady, no body forces, no viscous terms. Use Bernoulli eqn. to relate pressure & velocity. Force on a control volume

Incompressible flow = 

or  is the angle between  and 

For a wing section, 

= Can calculate lift using inviscid, incompressible flow if we know how much circulation to put in.

= Cannot calculate any drag without viscosity, except the induced drag due to finite aspect ratio.

= For airplanes at "usual" angles of attack at low speeds, most of the drag is induced drag, so such methods are useful.

= Away from flow boundaries, if  , inviscid flow is OK.