Governing Equations of Fluid Mechanics

#+CATEGORY: cfd

Governing Equations of Fluid Mechanics :: Fundamental Conservation Laws

The governing equations state a simple principle. Mass, momentum, and energy flow through a control volume. They do not appear or disappear. This note presents the equations in differential and integral form. It identifies the closure assumptions. It also flags which forms each subsequent wiki note uses.

Conservation laws derives the equations from the Reynolds transport theorem. The derivation leads to the differential form that every CFD code uses. Incompressible flow and Compressible flow branch from this tree.

Continuity Equation

The integral form holds for any control volume V bounded by surface S.

The differential (conservative) form is the one your solver marches forward:

The incompressible limit has constant density. This reduction yields divergence of u equals zero. It breaks the pressure-density coupling. See Incompressible flow.

Momentum Equation

Cauchy's equation appears before the constitutive relation enters:

The stress tensor is sigma. The body force per unit mass is f. It can be gravity or electromagnetic force. A Newtonian fluid satisfies this relation:

Substitution yields the Navier-Stokes equations. The convective term u dot nabla u creates non-linearity. This non-linearity drives the complexity. It also makes iterative solvers mandatory.

See Turbulence models for what happens when the viscous term cannot damp the non-linearity.

Energy Equation

Total energy per unit mass is e plus one-half u dot u. The internal energy is e. The conservative form is:

Constant thermo-physical properties decouple the energy equation from momentum. You can solve it after the momentum step. This works for incompressible flow with negligible viscous dissipation. Compressible flow requires simultaneous solution. See Compressible flow.

Constitutive Closure

The system has more unknowns than equations. A constitutive model closes the gap. The two common closures are:

1. Equation of state (EOS): p equals rho R T for ideal gases. Liquids use tabulated data or incompressible models. 2. Newtonian stress-strain: given above. Non-Newtonian fluids replace constant mu with a shear-rate-dependent function. Examples include power-law, Carreau, and Bingham models.

Ideal gases satisfy de equals cv dT and dh equals cp dT. Enthalpy h equals e plus p over rho. Incompressible liquids have constant properties. You can also use prescribed lookup tables.

System Count and Well-Posedness

Incompressible flow has four unknowns and four equations. The unknowns are u, v, w, and p. Temperature and transport scalars are additional.

Compressible flow has 5 equations and 5 unknowns. The conservative form provides 5 equations. The unknowns are rho, u, v, w, and p. Temperature follows from the EOS once you solve for the other variables.

Well-posedness requires the correct number of boundary conditions per characteristic direction. See Boundary conditions. Characteristic counting is trivial for incompressible flow but painful for supersonic compressible flow with chemistry.

Summary

• Continuity handles mass conservation
• Momentum expresses Newton's second law in continuum form
• Energy expresses the first law of thermodynamics
• EOS and constitutive law provide closure
• System size determines which solver family you must select.

References

• Papee, E. (1947). Navier-Stokes equations derivation.
• Batchelor, G. K. (2000). An Introduction to Fluid dynamics.
• Ferziger and Peric (2002). Computational Methods for Fluid dynamics.
• Versteeg and Malalasekera (2007). An Introduction to Computational Fluid dynamics.

See Also

• Conservation Laws
• Incompressible Flow
• Compressible Flow
• Turbulence Models
• Numerical Schemes
• Boundary Conditions