2.13 Incompressible flow

We can derive a set of equations for incompressible flow that includes:

  • mass conservation, Eq. (2.8 );
  • momentum conservation, Eq. (2.19 );
  • the material derivative for eqn, Eq. (2.26 );
  • the rate of deformation tensor, Eq. (2.33 );
  • the Newtonian fluid model, Eq. (2.41 ).

Combining Eq. (2.41 ) and Eq. (2.33 ) and using Eq. (2.36 ) for the deviatoric part of a tensor, gives an expression for stress:

 ☐☐☐ = ru + ruT 2tr(ru)I pI: 3 \relax \special {t4ht=
Substituting eqn into Eq. (2.19 ) and applying Eq. (2.26 ) gives an equation for momentum for a Newtonian fluid:
@---u h Ti @t + r ( uu) r ( ru) r (ru) 2- + 3 r(r u) = rp + b: \relax \special {t4ht=
The derivation of the equation above uses Eq. (2.35 ) and the identity eqn on page 84 .

An incompressible fluid exhibits eqn constant over time, i.e. for moving volumes of fluid eqn Combining the material derivative Eq. (2.14 ) and mass conservation Eq. (2.8 ) gives eqn which results in the incompressibility condition

r u = 0: \relax \special {t4ht=

A homogeneous, incompressible material exhibits eqn = constant uniformly throughout the entire fluid. With that assumption, Eq. (2.45 ) can be written as

@u @t-+ r (uu) r ( ru) (r ) (ru) = rp + b: \relax \special {t4ht=
This is the momentum equation for a homogeneous, incompressible Newtonian fluid. It includes: kinematic viscosity eqn in SI units of eqn; and eqn represents kinematic pressure, i.e. divided by eqn, in SI units of eqn.

The identity eqn and the incompressibility Eq. (2.46 ) yield the terms in Eq. (2.47 ).

Pressure equation

Mass and momentum conservation, represented by Eq. (2.46 ) and Eq. (2.47 ) respectively, provide two equations — one scalar, one vector — for two fields, eqn and eqn. However, Eq. (2.46 ) cannot be solved in its own right since it provides only one equation for vector eqn, containing 3 components.

A scalar equation, including both eqn and eqn, can be derived by taking the divergence of Eq. (2.47 ) and eliminating terms by substituting Eq. (2.46 ), noting that eqn. For eqn and eqn that are constant and uniform, the equation is

 2 r p + r [r (uu)] = 0: \relax \special {t4ht=
This pressure equation can replace Eq. (2.46) to provide a pair of equations, with Eq. (2.47 ), for both eqn and eqn. Chapter 5 describes the algorithms used in the finite volume method which couple the two equations using a modified form of Eq. (2.48 ).
Notes on CFD: General Principles - 2.13 Incompressible flow