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The Energy Equation for Incompressible Flow

Key Takeaways

  • All moving fluids have some kinetic and potential energy that determines their flow behavior.

  • The energy equation for incompressible inviscid laminar steady flow is better known as Bernoulli’s equation, although the two are not strictly the same.

  • Bernoulli’s equation makes a statement about the kinetic energy density along a streamline and is a universal relation for steady laminar incompressible flows.

Bernoulli’s equation

Anytime you do work on a fluid, you provide it with some kinetic energy and cause the fluid to begin flowing. The total energy of a fluid can be derived from the Navier-Stokes equations as long as all forces acting on the fluid are known. As long as the fluid flow is laminar, steady, incompressible, and inviscid, we can summarize the flow behavior in terms of a simple relationship known as Bernoulli’s equation.

The energy equation for incompressible flow is equivalent to Bernoulli’s equation and is a universal relationship. Bernoulli’s equation is very useful from a design perspective, as it can be used to track constant flow rate contours (streamlines) throughout a system. In particular, streamlines can be extracted from CFD simulations and easily used to track flow throughout a system. Bernoulli’s equation, when applied to one streamline, can also be used to understand flow behavior along any other streamline. Let’s take a closer look at this equation.

An Energy Equation for Incompressible Flow

To get to an energy equation for incompressible flow, the typical derivation starts from Euler’s equation of motion for fluid flow. Under some specific conditions, it is possible to arrive at a simple equation that describes the energy of the fluid, known as Bernoulli’s equation. Starting from Euler’s equations is much easier than starting from the full Navier-Stokes equation.

For incompressible steady laminar inviscid flows, Bernoulli’s equation is:

Energy equation for incompressible flow

This equation relates the flow velocity u to the driving pressure P and the potential energy associated with any other time-independent conservative forces acting on the fluid. The SI (mks) units of this equation are J/kg, meaning the equation expresses a kinetic energy per unit mass. This equation could be multiplied by the fluid density to get a kinetic energy per unit volume.

The above equation is universal, as it tells you the kinetic energy along a streamline for any steady incompressible inviscid laminar flow. In the case where viscosity is non-negligible, or when driving forces are unsteady, the above equation will no longer apply, and we have special cases of Bernoulli’s equation that should be derived from the Navier-Stokes equations or from CFD simulations.

Can Energy Be Unsteady?

In short, the answer is “yes,” but this would mean mechanical energy was being given to the fluid, or the fluid was losing its mechanical energy during flow. It could also mean some mechanical energy was being transformed into another type of energy (e.g., thermal energy) and be lost from the system. In this case, Bernoulli’s equation in the form shown above would no longer hold. There are two broad notable cases that can be discussed where we would have a different form of Bernoulli’s equation where fluid may be unsteady.

Time-Dependent Potential Energy and Forces

If the potential energy governing fluid flow were unsteady, then the kinetic energy could also be unsteady. Whether the kinetic energy compensates for fluctuations in potential energy due to some characteristic in the system, or whether energy is totally un-conserved, depends on the nature of the potential energy fluctuation. This is one aspect of fluid flow that is best investigated using time-dependent CFD simulations.

Compressible Flows

By definition, a compressible flow will not be steady or homogeneous; this is because the flow density must change somewhere along the streamlines at some point in time. When the flow is compressible, the energy of the fluid may still be conserved if the flow is slow enough. If the compression of the flow is very slow such that its temperature basically remains constant, then the energy of the moving fluid can be regarded as constant. We could then conceivably derive a compressible version of Bernoulli’s equation that accounts for isothermal compression.

In adiabatic compression (e.g., in a gas), the temperature of the fluid will change during compression/decompression and heat will be exchanged with the surrounding environment. Some mechanical energy may be lost as heat to the surroundings in compressible flow. In the case where a fluid is totally insulated from its surroundings, then the fluid’s energy would be conserved and all compression would be adiabatic. For purely adiabatic compressible flows, Bernoulli’s equation can be rewritten in terms of the fluid’s adiabatic index:

Bernoulli’s equation compressible flow

Viscous Flow

By definition, viscous forces are non-conservative, meaning they do not conserve mechanical energy. Viscosity is the reason flows will lose their kinetic energy as soon as the driving force is removed from a system and viscosity is allowed to dominate flow behavior. Viscosity is like friction: it will convert mechanical energy into heat. In this way, mechanical energy is not conserved but total energy is conserved once we account for heat generation in the system.

Viscous flow in honey

Viscous flows will experience a loss of mechanical energy because viscous forces are non-conservative.

These more complex flows, such as compressible flows with time-dependent forces, will have an energy equation that does not match Bernoulli’s equation and which may not be constant in time. CFD simulations can be used in these systems to examine flow behavior, and the resulting numerical simulation data can be used to calculate an energy model using regression.

When you need to investigate an energy equation for incompressible flow or more complex compressible viscous flows, 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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