General Single-Phase Flow Theory
The Single-Phase Fluid Flow interfaces are based on the Navier–Stokes equations, which in their most general form read
(3-4)
(3-5)
(3-6)
where
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ρ is the density (SI unit: kg/m3)
•
u is the velocity vector (SI unit: m/s)
•
p is pressure (SI unit: Pa)
•
I is the identity matrix (unitless)
•
K is the viscous stress tensor (SI unit: Pa)
•
F is the volume force vector (SI unit: N/m3)
•
Cp is the specific heat capacity at constant pressure (SI unit: J/(kg·K))
•
T is the absolute temperature (SI unit: K)
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q is the heat flux vector (SI unit: W/m2)
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Q contains the heat sources (SI unit: W/m3)
•
S is the strain-rate tensor:
The operation “:” denotes a contraction between tensors defined by
(3-7)
This is sometimes referred to as the double dot product.
Equation 3-4 is the continuity equation and represents conservation of mass. Equation 3-5 is a vector equation which represents conservation of momentum. Equation 3-6 describes the conservation of energy, formulated in terms of temperature. This is an intuitive formulation that facilitates boundary condition specifications.
To close the equation system, Equation 3-4 through Equation 3-6, constitutive relations are needed.
For a Newtonian fluid, which has a linear relationship between stress and strain, Stokes (Ref. 1) deduced the following expression:
(3-8)
The dynamic viscosity, μ (SI unit: Pa·s), for a Newtonian fluid is allowed to depend on the thermodynamic state but not on the velocity field. All gases and many liquids can be considered Newtonian.
For an inelastic non-Newtonian fluid, the relationship between stress and strain rate is nonlinear, and an apparent viscosity is introduced instead of the dynamic viscosity. Examples of non-Newtonian fluids are honey, mud, blood, liquid metals, and most polymer solutions.
With the Polymer Flow Module, you can model flows of non-Newtonian fluids using the predefined constitutive models. The following models which describe the stress-strain relationship for non-Newtonian fluids are available: Power law, Carreau, Carreau–Yasuda, Cross, Cross–Williamson, Bingham–Papanastasiou, Herschel–Bulkley–Papanastasiou, Casson–Papanastasiou, DeKee–Turcotte–Papanastasiou, Robertson–Stiff–Papanastasiou, Ellis, and Houska thixotropy.
In theory, the same equations describe both laminar and turbulent flows. In practice, however, the mesh resolution required to simulate turbulence with the Laminar Flow interface makes such an approach impractical.
There are several books where derivations of the Navier–Stokes equations and detailed explanations of concepts such as Newtonian fluids can be found. See, for example, the classical text by Batchelor (Ref. 3) and the more recent work by Panton (Ref. 4).
Many applications describe isothermal flows for which Equation 3-6 is decoupled from Equation 3-4 and Equation 3-5.
2D Axisymmetric Formulations
A 2D axisymmetric formulation of Equation 3-4 and Equation 3-5 requires to be zero. That is, there must be no gradients in the azimuthal direction. A common additional assumption is, however, that . In such cases, the -equation can be removed from Equation 3-5. The resulting system of equations is both easier to converge and computationally less expensive compared to retaining the -equation. The default 2D axisymmetric formulation of Equation 3-4 and Equation 3-5 therefore assumes that