Bernoulli's Equation Derivation From the Navier-Stokes Equation
Bernoulli’s equation is a statement of conservation of energy in incompressible laminar flows without surface friction effects like drag.
Some simple assumptions are needed to derive Bernoulli’s equation.
The Bernoulli’s equation derivation from Navier-Stokes is simple and relies on applying linearization.
Bernoulli’s principle is a theoretical relation describing fluid flow behavior for incompressible laminar flows. In particular, Bernoulli’s equation relates the flow parameters along a given streamline to the potential energy in the system and the pressure that drives laminar flow. In effect, it is a statement about conservation of energy in incompressible inviscid laminar flow under some simple assumptions. These assumptions can be very nearly realized in many systems, giving Bernoulli’s equation wide applicability.
Although Bernoulli’s equation is system-specific, the equation is universal in that its derivation from the Navier-Stokes equation does not rely on the application of boundary conditions or specific geometries. In this article, we’ll present the Bernoulli’s equation derivation from the Navier-Stokes equation of motion. Later, we’ll briefly discuss how Bernoulli’s equation extends to compressible flows and slightly unsteady flows.
Bernoulli’s Principle Derivation
One does not perform the Bernoulli’s equation derivation from the Navier-Stokes equation of motion. Instead, we start from Euler’s equation for incompressible flow. Bernoulli’s equation is an acceptable result that is easily derived from Euler’s equations, which is just a quasi-linearized form of the full Navier-Stokes equation.
As Bernoulli’s equation is basically a statement on the conservation of energy for the fluid, we start with a few assumptions:
Conservative forces: All vector forces acting on the fluid are considered to be conservative. This means they can be calculated from the gradient of a scalar potential function.
Inviscid flow: The fluid viscosity is assumed to be zero or so small that it is negligible compared to any externally applied forces acting on the fluid. This follows naturally from the first assumption because viscous drag is a nonconservative force.
No turbulence: The absence of turbulence in the system also follows from the first two assumptions, as a zero viscosity term will eliminate the convective terms in the Navier-Stokes equation of motion.
Incompressibility: The simplest form of Bernoulli’s equation treats incompressible flows. There is a more complex form that treats compressible flows, as we’ll see below.
Steady flow: Bernoulli’s principle is normally applied only to steady flow, although it can be applied to flows with small fluctuations by keeping time derivative terms and expanding the applied forces with some fluctuations (shown below).
First, we begin with the quasilinear Euler equation in 3D:
Next, we apply the assumptions in the above list to rewrite the gradient terms:
If we now expand the flow rate dot product-gradient term in the Euler equation, we will have terms that vanish thanks to the irrotational assumption shown above. We can also substitute the scalar potentials into the equation. The Euler equation then reduces to:
Now we can collect everything on the LHS of the above equation and pull off the gradient operator from the equation to get the following result:
Applying the steady flow assumption reduces the time derivative term to zero. Here, we have a gradient of a scalar quantity being equal to zero, thus we pull off the gradient operator to arrive at Bernoulli’s equation:
Extension to Compressible Flow
In compressible flow, knowledge of the relationship between fluid density and pressure is needed to determine a form of Bernoulli’s equation for compressible fluids:
If we assume fluid deformation to be purely adiabatic, we would arrive at the following relationship using integration by parts:
Here, 𝛾 is the adiabatic index, which depends on the molecular structure of the fluid. The subscripted density and pressure quantities are stagnation (static) terms. The above relationship allows Bernoulli’s equation to be extended to laminar flows involving gases.
Extension to Unsteady Flow
Extending Bernoulli’s principle to unsteady flows requires rewriting the potential energy in terms of small fluctuations about a mean as follows:
If we go back to the gradient equation above, apply this mean-variance transformation, and remove the gradient operator, we have the following equation:
After integrating away the gradient operator, we have a set of constant terms plus a set of time-dependent terms; the latter set of terms is equal to zero. Therefore, we arrive at the following relationship between the potential energy fluctuations and the flow rate fluctuations:
Note that U is a potential energy function per unit mass. Applying the gradient operator to the above equation gives the definition of acceleration. This tells us that any velocity fluctuations are described entirely by potential energy fluctuations as long as all other assumptions in Bernoulli’s equation are satisfied.
The compressible relationship shown above can also be applied when deriving the compressible form of Bernoulli’s equation, however, this will be left as an exercise for the reader.
Determine the Flow Rate Profile in CFD Simulations
The primary task that must be completed before using Bernoulli’s equation in any of its above forms is to determine the flow rate profile for your system. In particular, your CFD simulation software should allow you to visualize streamlines that follow Bernoulli’s equation. From here, you can infer how flow behavior will change when system or fluid parameters are changed, which may eliminate the need to run multiple simulations to understand laminar flows.
When you need to investigate flows and verify the standard application of Bernoulli’s principle, you can build and run high-accuracy CFD simulations using Omnis from Cadence. Modern numerical approaches used in aerodynamics simulations, turbulent and laminar flow simulations, reduced fluid flow models, and much more can be implemented in Cadence’s simulation tools.
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