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Energy Equations: The Navier-Stokes Method of Analysis

Key Takeaways

  • The Navier-Stokes equations follow the principle of conservation of the energy, momentum, and mass of a fluid flow.

  • The energy equation of the Navier-Stokes system follows the energy conservation law, which equates the total energy of a system to the sum of work and heat added to the system. 

  • In CFD simulations, the Navier-Stokes energy equation provides the basic explanation of energy associated with the flow behavior. 

Fluid mechanics

Let us start by taking the example of a fluid flowing through a pipe. One must analyze the mass, momentum, and energy associated with the flow. The simplification of these factors is the clue to understanding flow physics in different fluid systems. One way to easily do this is with the help of the Navier-Stokes equations.

In fluid dynamics, the Navier-Stokes equations deal with the conservation of energy, momentum, and mass of a fluid flow. These are also the governing equations in CFD (computational fluid dynamics). In this article, we will touch upon the different Navier-Stokes equations and discuss the energy equation in Navier-Stokes CFD analysis for both compressible and incompressible fluid.

The Navier-Stokes Equations

The Navier-Stokes equations represent the partial differential equations that explain the flow phenomenon of a viscous, incompressible fluid. They are based on the principle of conservation of mass, momentum, and energy. These are the governing principles of fluid in motion and can be widely used during vehicle design, pipe flow modeling, airfoil design, and so on. Following the above conservation principles, the Navier-Stokes equations encompass the following major equations:

  1. Continuity equation
    Within a controlled volume, the mass of the flow remains constant with time. This is the law of conservation of mass and can be expressed as:

 Continuity equation

V is the velocity and ρ is the density.

  1. Newton’s second law of motion (F=ma) or the momentum equation
    The change of momentum over time is equal to the sum of force acting on the system. This Navier-Stokes equation is reflective of the law of conservation of momentum. The momentum equation can be expressed as: 

 Momentum equation

τ is the viscous stress and f represents the force per unit mass.

  1. First law of thermodynamics or the energy equation
    Similarly, the first law of thermodynamics reflects the Navier-Stokes energy equation. The energy conservation law states that the total energy of the system equals the sum of work and heat added to the system. One way to express this is the following equation. 

    Energy equation

Note that, h is the enthalpy, k is the thermal conductivity of the fluid, and Φ represents dissipation. In the above equation:

I: Energy changes with time

II: Convection term

III: Pressure work 

IV: Diffusion term

V: Dissipation term

These are the three fundamental Navier-Stokes equations for compressible fluids. 

Navier-Stokes Energy Equations for Incompressible Flow

The energy equation discussed in equation 3 is one way the Navier-Stokes equations can be expressed for compressible flow. In the case of the ideal fluid flow, the above Navier-Stokes energy equation can be further simplified in terms of enthalpy. We know that:

, where Cp is the specific heat.

Equation 3 now simplifies in terms of enthalpy as:

 Simplified energy equation

The relation between density, pressure, and temperature in a compressible flow is provided by an equation of state, which is the following equation, where R is the gas constant:

p=⍴RT

However, for incompressible flow, the equation of state also does not apply. The fluid density is constant over time, resulting in decoupling from pressure and velocity. Given the relationship to pressure is not evident in the Navier-Stokes equations, the solution for the given flow field becomes much more complex.

One way to solve the energy equation for incompressible flow is to work with Euler equations, which help relate the velocity and pressure related to the flow. This relation can then be introduced to density to identify the energy in the flow system. 

Numerical Simplification of the Energy Equation: Navier-Stokes Analysis

The theoretical analysis of a flow model using CFD requires a proper understanding of flow parameters and behaviors. The Navier-Stokes energy equations are the basis of this understanding, as they simplify the flow behavior analysis for different fluid systems. A numerical approach to these equations is the key to simulation accuracy when dealing with complex systems. CFD simulation tools can help engineers solve the momentum, mass, and Navier-Stokes energy equations.

CFD software from Cadence can generate detailed numerical meshes and design simulations that can ideally represent the flow factors, facilitating the analysis of their effects on the system design. High-fidelity simulation allows engineers and designers to optimize the fluid system design and aim for efficiency.

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