Bernoulli’s principle is a universal relation describing flow behavior for ideal fluids.
Some common applications of Bernoulli’s principle are its use to explain flow behavior in simple systems.
More complex flow behavior can be explained with modified versions of Bernoulli’s principle.
Some relations from fluid dynamics are universal, and one of these is Bernoulli’s principle. This simple relationship defines a range of flow behavior for ideal fluids, but many real systems can be approximated in such a way that Bernoulli’s principle is relevant. To see where applications of Bernoulli’s principle can be treated with the standard form, we’ll look at a few common fluid flow situations that illustrate the applicability of this universal relation.
Are There Applications of Bernoulli’s Principle?
Systems engineers do not necessarily use Bernoulli’s principle as the basis for designing systems that rely on fluid flow. Bernoulli’s principle is an explanatory tool that describes why fluid behavior occurs in certain ways—it is not really a design tool. That being said, there are some applications or examples of fluid flow that can be best explained and understood using Bernoulli’s principle. As we’ll see in the next section, Bernoulli’s principle is something of a universal relation involving certain types of flows.
What Is Bernoulli’s Principle?
Bernoulli’s principle is essentially a statement regarding the conservation of energy in a flowing fluid, and it defines the conservation of mechanical energy for all streamlines that make up the flow. There is a simple form of Bernoulli’s equation that can be derived from Euler’s equations describing certain types of flows:
Steady flow: Bernoulli’s principle only applies to steady flows, as unsteady flows would require the addition or dissipation of energy in the fluid by an external force.
Incompressible: The fluid density appears in Bernoulli’s principle, but the fluid density is assumed constant for all streamlines. Note that this does not mean the fluid energy will be the same everywhere; this will depend on the velocity field.
Irrotational: The flow is defined as being irrotational everywhere. This is equivalent to stating that there are no convective forces acting on the flowing fluid, i.e., the flow is in the laminar regime.
Inviscid: The fluid is assumed to be inviscid or approximately inviscid along all streamlines. In other words, there is no frictional force that would cause mechanical energy to be lost to heat.
Conservative forces: The counterpart to the previous point is that all forces acting on the fluid are conservative. This is not possible in reality, although in low Reynolds number flows we can approximate that only conservative forces are dominant.
The simplest form of Bernoulli’s principle for these types of flows defines a constitutive relation for any streamline in pressure-driven fluid flow:
Simplest form of Bernoulli’s principle for incompressible, irrotational, inviscid flows.
With this in mind, we can look at a few common fluid flow situations that can be explained using Bernoulli’s principle.
Lift in Aerodynamics
In aerodynamics, fluid flow across an airfoil will generate lift due to a vertical pressure gradient along the cross section of the wing. The pressure gradient in the vertical direction can be predicted using Bernoulli’s equation from the flow streamlines. Streamlines are commonly visualized along an airfoil, which allows the pressure gradient and lift to be predicted.
From the example streamline map shown below, we can see clearly that airflow is faster along the top surface of the airfoil, thus pressure will be higher on the bottom surface, creating lift. We can also see where flow separation and vortical flow occur along the back side of the wing, which would be expected at high Reynolds numbers.
Example streamline map along an airfoil (Image source).
Flow Through Nozzles and Channels
Low Reynolds number flows being directed through nozzles or channels can be described using Bernoulli’s principle. In particular, the change in flow rate as the system dimensions change can be predicted using Bernoulli’s equation. This effectively explains the Venturi effect, where a pressure-driven flow rate will change in accordance with conservation of energy.
Fluid Flow Measurements
A related application of Bernoulli’s principle is fluid flow rate measurements. The flow rate of a fluid can be measured by taking advantage of the Venturi effect. As fluid flows into an orifice plate with a small aperture of known diameter, the reduction in diameter will cause an increase in the fluid flow speed. Based on this measurement inside the channel, the flow rate outside the orifice plate can be calculated using Bernoulli’s principle.
Drag is present in aerodynamics, but it is also present generally whenever a gas or liquid flows across an object. One example is in watercraft, where streamline can be used to predict the drag force, or additionally the propulsive force such as in a sailboat (see below). Another prominent area of application is to explain and visualize drag along the body of an automobile, something which can be verified in a wind tunnel with controlled airflow.
Streamlines along watercraft.
Applications of Alternative Forms of Bernoulli’s Principle
The form of Bernoulli’s principle shown above has limited applicability. In real systems, the fluid may be compressible or the fluid may be subject to time-dependent forces. In these cases, a modified form of Bernoulli’s equation is needed to describe flow behavior along streamlines. These cases lead to some additional broad applications of Bernoulli’s principle:
- Slowly-varying flows, where the applied forces or flow parameters have slow variations.
- Isothermal compression of inviscid flowing gases, where compression occurs slow enough that total thermal energy is conserved during compression.
- Fluid flow in mechanical waves, which is used to explain harmonic motion of a fluid in acoustics and water waves.
These expanded applications of Bernoulli’s principle require a deviation from the typical form shown above, and the new form of Bernoulli’s principle may not have an analytical description. CFD simulations are most useful for these applications, as they allow a system-specific form of Bernoulli’s equation to be derived from simulation data.
If you’re working with a more advanced system involving an application of Bernoulli’s principle, you can investigate flow behavior in your design with the meshing tools in Pointwise and the complete set of fluid dynamics analysis and simulation tools in Omnis 3D Solver from Cadence. These two applications give systems designers everything they need to build and run CFD simulations with modern numerical approaches.