1.4 Bernoulli's equation

Bernoulli's equation is a cornerstone of fluid dynamics that links pressure, velocity, and elevation along a streamline. It comes from energy conservation principles applied to steady, incompressible flow. This tool helps engineers analyze and design aircraft components ranging from wings to engine inlets.

While invaluable, Bernoulli's equation has real limitations. It assumes inviscid flow and doesn't account for compressibility or energy losses. Understanding these constraints is just as important as knowing the equation itself.

Bernoulli's equation derivation

Bernoulli's equation connects three quantities along a streamline: pressure, velocity, and elevation. The derivation starts from the conservation of energy and relies on a few key assumptions about the flow. Getting comfortable with these assumptions tells you exactly when the equation applies and when it doesn't.

Steady flow assumption

The flow properties (pressure, velocity, density) at any given point don't change with time. This lets you use streamlines, which are curves tangent to the velocity vector at every point in the flow field. Because nothing varies in time, all the time-dependent terms in the governing equations drop out, which greatly simplifies the math.

Incompressible flow assumption

The fluid density $\rho$ stays constant everywhere in the flow. This is a good approximation for low-speed flows where the Mach number (the ratio of flow velocity to the local speed of sound) is below about 0.3. With constant density, the conservation of mass simplifies to:

$\nabla \cdot \vec{V} = 0$

Above Mach 0.3, density changes become significant and you need compressible flow equations instead.

Conservation of energy principle

Energy can't be created or destroyed, only converted between forms. In a flowing fluid, the total mechanical energy per unit volume has three components:

• Kinetic energy: $\frac{1}{2}\rho V^2$
• Potential energy: $\rho g z$
• Pressure energy (flow work): $p$

When you apply conservation of energy along a streamline for steady, incompressible, inviscid flow, these three terms must sum to a constant. That's Bernoulli's equation.

Pressure, velocity, and elevation terms

The full equation between any two points on the same streamline is:

$p_1 + \frac{1}{2}\rho V_1^2 + \rho g z_1 = p_2 + \frac{1}{2}\rho V_2^2 + \rho g z_2$

The key takeaway: if velocity increases, pressure must decrease (and vice versa), assuming elevation stays roughly constant. Each term has units of pressure (Pa), and together they represent the total mechanical energy per unit volume of the fluid.

Bernoulli's equation applications

Pitot tubes for airspeed measurement

A pitot tube measures airspeed by comparing two pressures. At the tube's forward-facing inlet, the flow stagnates, producing the stagnation pressure $p_0$. Side holes measure the static pressure $p_s$ of the undisturbed flow. Bernoulli's equation connects them:

$p_0 = p_s + \frac{1}{2}\rho V^2$

Rearranging to solve for airspeed:

$V = \sqrt{\frac{2(p_0 - p_s)}{\rho}}$

So by measuring the pressure difference $p_0 - p_s$ and knowing the air density, you get the airspeed directly. This is the principle behind the pitot-static systems on virtually every aircraft.

Venturi effect in carburetors

Carburetors exploit the Venturi effect to mix fuel with air. As air enters a converging section of tubing, the cross-sectional area decreases, so the air speeds up. By Bernoulli's equation, that higher velocity means lower pressure at the narrowest point (the throat). This low-pressure region draws fuel from a reservoir (the float chamber) into the airstream, atomizing it for combustion.

Lift generation on airfoils

Bernoulli's equation helps explain part of how airfoils generate lift. Air flowing over the curved upper surface of a wing travels faster than air along the flatter lower surface. Faster flow on top means lower pressure on top. The resulting pressure difference between the upper and lower surfaces produces a net upward force: lift.

Pressure drops in pipes and ducts

For internal flows, Bernoulli's equation helps estimate how pressure changes between two points in a pipe or duct. Changes in cross-sectional area or elevation cause corresponding changes in velocity and pressure. Applying the equation between an upstream and downstream point lets you calculate the expected pressure drop, which is useful for sizing pipes and selecting pumps. In practice, you'll also need to account for frictional losses (more on that in the limitations section).

Limitations of Bernoulli's equation

Validity for inviscid flows

Bernoulli's equation assumes inviscid flow, meaning no viscous forces (friction) act on the fluid. Every real fluid has viscosity, which creates boundary layers near solid surfaces and dissipates energy. The inviscid assumption works reasonably well for high-Reynolds-number flows in regions away from solid boundaries, but it breaks down inside boundary layers and in separated flow regions.

Inapplicability to compressible flows

The standard form of Bernoulli's equation assumes constant density. Once the Mach number exceeds roughly 0.3, compressibility effects become significant and density can no longer be treated as constant. For example, at Mach 0.5 the density change is already around 13%. At higher speeds, you need the compressible Bernoulli equation or isentropic flow relations instead.

Inability to account for energy losses

The equation conserves total mechanical energy along a streamline, so it has no way to represent energy lost to friction, turbulence, or other dissipative processes. In real flows, these losses cause pressure drops beyond what Bernoulli predicts. To handle this, engineers add loss terms. For pipe flow, the Darcy-Weisbach equation provides a way to estimate frictional head loss and incorporate it into the energy balance.

Bernoulli's equation vs Euler's equation

These two equations are both derived from conservation of momentum for inviscid flow, but they differ in generality and form.

Assumptions and simplifications

• Bernoulli's equation assumes steady, incompressible, inviscid flow along a streamline, with no shaft work or heat transfer. It's a scalar equation relating pressure, velocity, and elevation.
• Euler's equation is more general. It can handle unsteady and compressible inviscid flows. It's a vector equation that includes the pressure gradient and body forces, and it applies throughout the flow field, not just along a streamline.

You can think of Bernoulli's equation as a special, integrated form of Euler's equation under more restrictive assumptions.

Applicability to different flow scenarios

For flows with significant viscous effects or boundary layers, neither equation is sufficient. You need the Navier-Stokes equations, which add viscous stress terms to Euler's equation.

Bernoulli's equation in aerodynamic design

Airfoil shape optimization

Bernoulli's equation guides the relationship between airfoil geometry and the resulting pressure distribution. By shaping the airfoil contour, designers control how velocity (and therefore pressure) varies along the surface. Techniques like inverse design methods start with a desired pressure distribution and work backward to find the airfoil shape that produces it. Optimization algorithms can then search for shapes that maximize lift-to-drag ratio for specific flight conditions.

Wind tunnel testing and validation

Wind tunnel tests validate whether real flow behavior matches the predictions from Bernoulli-based analysis. Scale models of wings, fuselages, or other components are instrumented with pressure taps to measure surface pressure distributions, lift, and drag. Comparing these measurements against theoretical predictions reveals where the inviscid, incompressible assumptions hold and where viscous or compressible effects require more sophisticated modeling.

Computational fluid dynamics simulations

CFD simulations solve the full Navier-Stokes equations numerically, capturing viscous and compressible effects that Bernoulli's equation can't. These simulations provide detailed velocity, pressure, and temperature fields around complex geometries. Bernoulli's equation still plays a role here: it provides a quick sanity check on CFD results and helps engineers interpret pressure-velocity relationships in the computed flow fields.

Bernoulli's equation extensions

The standard form of Bernoulli's equation is limited to steady, incompressible, inviscid flow. Several extensions broaden its applicability.

Compressible flow corrections

For compressible flows, density varies and you can't treat $\rho$ as constant. The compressible Bernoulli equation is written in terms of enthalpy rather than pressure:

$h_0 = h + \frac{1}{2}V^2$

Here $h_0$ is the stagnation (total) enthalpy and $h$ is the static enthalpy. This form accounts for density changes and can be extended further to handle shock waves and other compressible phenomena. For an ideal gas undergoing isentropic flow, enthalpy relates directly to temperature, connecting this equation to the isentropic flow relations you'll encounter later.

Viscous flow modifications

To handle real viscous flows, a head loss term $h_L$ is added to represent energy dissipated by friction:

$p_1 + \frac{1}{2}\rho V_1^2 + \rho g z_1 = p_2 + \frac{1}{2}\rho V_2^2 + \rho g z_2 + \rho g h_L$

The head loss $h_L$ can be estimated using empirical correlations (like the Darcy-Weisbach equation for pipe flow) or derived from the Navier-Stokes equations for specific geometries. This modified form is sometimes called the extended Bernoulli equation or the energy equation for pipe flow.

Unsteady flow adaptations

When flow properties change with time, the unsteady Bernoulli equation introduces a time-dependent term involving the velocity potential $\phi$:

$\frac{\partial \phi}{\partial t} + \frac{p}{\rho} + \frac{1}{2}V^2 + gz = C(t)$

where $C(t)$ is a function of time only (constant across space at any given instant). This form is useful for analyzing transient events like flow startup in a pipe, water hammer, or an airfoil's response to gusts and turbulence.