The Bernoulli Equation

In the US, many students' first introduction to aerodynamic principles happens in basic physics or natural sciences coursework with a lesson on the "Bernoulli effect." Unfortunately, for a lot of people this is also the end of their education in aerodynamics, leading to a lot of incorrect understanding, weird theories, and gross oversimplification—like the incorrect idea that a fully streamlined body with no flow separation will have no pressure drag (it still does).
 
You probably recall the gist of it: as the velocity of a stream of air increases, its static pressure is reduced. This qualitative relationship is always true of low speed flows with no energy addition, since (total) pressure is a measure of the (total) energy in a volume of air and, because energy cannot be created or destroyed spontaneously, when more of that energy is used in the kinetic energy of the flow there must be less internal energy (static pressure) available.

This is one explanation of how wings make lift, although it has nothing to do with the "equal transit theorem" that many students are taught (incorrectly).

But the Bernoulli "effect" is actually a lot more specific than that, and it only applies in certain, narrow circumstances—when we can make assumptions about the flow that may or may not be true. Let's see what those are.
 
Bernoulli Equation
 
First off, the label "Bernoulli effect" or "Bernoulli principle" isn’t quite correct. More accurately, we should call it the "Bernoulli description" but in aerospace engineering we refer to it always as the "Bernoulli equation." Daniel Bernoulli was an 18^th-century Swiss mathematician who came from a family of scientists; one of his achievements was formulating an equation that describes an attribute of fluid flows under certain conditions that underpins flow models and CFD in aerospace engineering to this day. The Bernoulli equation describes the relationship between static pressure and dynamic pressure along a streamline:

…where k is some constant total pressure. This equation only applies if we make certain assumptions; specifically, five of them:
 
1) inviscid flow
2) incompressible flow
3) irrotational flow
4) negligible body forces
5) steady flow
 
Only when these five conditions are met does the Bernoulli equation apply. This is the first thing most people get wrong about aerodynamics, assuming that this relationship holds anywhere in a flow (incorrectly assuming that total pressure is constant everywhere). It does not, and the first reason it doesn't is because anywhere we have flow past a body or surface, there will be friction forces acting on it—so, we cannot assume it is inviscid. We can only neglect viscosity sufficiently far from a body or wall, outside the boundary layer that develops along every body submerged in moving air.
 
The Bernoulli equation also does not apply to dense fluids like water because their high mass per volume leads to non-negligible gravitational force (and introduces changes in potential energy, which is not accounted for in the equation). However, air has sufficiently low density (even when it contains water vapor, like clouds) that we can usually approximate the effect of gravity on it as nonexistent. This applies to small volumes only; if we take something like the hydrostatic equation and a large column of air (which describes the physical mechanism that results in the atmospheric pressure we feel at ground level), gravity is of course not negligible. But looking at a sufficiently small volume of air—say, the air immediately surrounding your car—the assumption of no body forces is generally valid.
 
We can divide flow conditions over time into steady and unsteady or transient classifications. The latter mean flow conditions change over time while the former means the properties of the flow are constant through time. Real flows are inherently subject to unsteady phenomena like turbulence, which means that no flow in the real world is truly steady. However, we can approximate a lot of flow problems as steady if the time interval under consideration is long enough, taking average properties as characteristic of the flow. This does not apply all the time, but when it does the Bernoulli equation holds.
 
That's three out of five; what about the last two? To explain these, we have to delve into vector math and how we use it to describe properties of the flow—in particular, the velocity field.
 
Velocity
 
Velocity is a vector, a composite of elements that define its magnitude and direction. Vectors can be written in a variety of formats but it is usual when representing velocity to break it into components along the three physical dimensions using the letters,

To make operations easier to perform, you will frequently see these vectors transposed, sometimes with a "T" (for "transpose") superscript indicator:

The gradient operator (sometimes called "grad" or "nabla") is another vector—but different from ordinary vectors, the elements of this one perform an operation by taking the partial derivative of another vector or function:

If a velocity field is given by some function of spatial coordinates (x, y, z),

then the gradient of the velocity is

This tells us the change in velocity with change in position. But this is not the only operation we can perform using the gradient operator. If we apply it using other vector relations, it can tell us two important things about the flow—and the validity of the Bernoulli equation.
 
Divergence
 
In vector math, there are different types of multiplication or products (this arises from the definition of vector subspaces and something called "closure properties," which allows for mathematical operations to be defined any way we want so long as the closure properties are not violated). "Divergence" is defined by the inner product (or dot product) of the gradient operator and the velocity:

This gives a scalar quantity with units s^-1 or Hz (hertz).
 
Divergence is an important parameter of the flow because it tells us whether the flow is compressible or incompressible. If the divergence is zero,

then the volume of a fluid element of mass dm does not change as it traverses its path along the flow. No volume change of a constant-mass element means constant density, ρ = c, and the flow is incompressible—one of our two remaining criteria for application of the Bernoulli equation.
 
Curl
 
The "curl" of the velocity is given by another type of multiplication, this time the cross product of the gradient operator and the velocity. This type of product outputs another vector orthogonal to both (you may be familiar with torque, which is the cross product of an applied force and a moment arm):

This gives a vector with units Hz.
 
As you might guess, this tells us something about rotation in the flow—but be careful! This is not vorticity, which is rotation in the velocity field, but the change in orientation of the body axes of any infinitesimal element dm with respect to world coordinates. If the curl is zero,

then the orientation of any fluid element dm does not change as it traverses its path along the flow and the flow is irrotational—satisfying the last criterion for application of the Bernoulli equation.
 
Velocity Field
 
Finally, if we apply the gradient operator to the velocity vector (the spatial function above will give the same result when decomposed), then our two vectors expand into another dimension—so two 1x3 vectors output a 3x3 matrix:

This gives the change in each component of velocity with change in spatial coordinates along the x, y, and z axes of whatever space we've defined. Notice that the center diagonal of this matrix contains the elements in the divergence of the flow, while the remaining elements, diagonally in the opposite direction, produce the components of the flow's curl.
 
Example
 
So, why is Bernoulli important? If we can show that a particular flow problem satisfies the five criteria necessary to be able to apply it, the Bernoulli equation lets us predict the velocity and static pressure anywhere in the flow. That's powerful! Aerospace engineers still use it today (as in the first image, a screen capture of an AVL simulation) to predict lift and aircraft performance with potential flow theory, another centuries-old body of mathematics. Let's look at a simple example to demonstrate.
 
Potential flow theory works by superposing one or more of several basic flows: uniform flow, sources, sinks, doublets (a combination of source and sink), and vortices. Defining the potential or stream functions of each of these, combining them in whatever way we want, and then taking the derivative gives the resulting velocity field. Say we have uniform horizontal flow from the left and a point vortex of strength Γ (gamma) at arbitrary coordinate (0, 0) in a plane. The resulting velocity field is given by:

Does this satisfy our criteria above for applying the Bernoulli equation? We're implicitly assuming steady flow, and we haven't included any gravitational field or viscosity model in our equations. What about incompressibility and irrotationality?
 
Drop the velocity field vector into the equations for divergence and curl, differentiate, and we find:

Since the divergence and curl of the velocity field are both zero, the flow is incompressible and irrotational. The Bernoulli equation applies, and the flow along any streamline has constant total pressure:

A similar process is used to model airfoil and wing shapes to predict their lift and induced drag, and these work very well—to a point. That point is the onset of stall, the angle of attack where the lift curve starts to deviate from inviscid theory because viscous effects begin to dominate. You can observe this on real wings with tufts:

This modified Clark-Y wing section achieved maximum C_l at angle of attack α = 14°, but you can see that already nearly half the wing's upper surface is in separated flow. As the separation bubble starts moving up the wing surface from the trailing edge and the wing approaches stall, inviscid models don't accurately predict performance! At this angle of attack, potential flow theory predicts a lift coefficient significantly higher than the actual maximum lift. The Bernoulli equation, which works just fine for predicting lift at low angles of attack, no longer applies because too much total pressure is lost to energy dissipation in the separated flow. (And we learn something important about the flow over wings: maximum lift does not occur with the wing supporting attached flow over its entire surface).
 
Important Lessons
 
1) The qualitative relationship between static pressure and dynamic pressure in low speed flows—as one goes up, the other must go down—is always true because energy must be conserved
2) The Bernoulli equation describes this relationship under a certain set of conditions in which total pressure along any streamline is constant
3) This inviscid, irrotational, incompressible model predicts lift forces very well up to stall
4) It cannot predict drag due to the same assumptions and it cannot account for momentum loss in real flows
5) So, the circumstances in which we can use the Bernoulli equation to model airflow are useful but narrow and limited—be careful!