Fundamental Equations

The conservation laws for a continuum can be expressed using an Eulerian approach or a Lagrangian approach. In the Eulerian approach, a given volume represents a portion of space where material can flow through. In the Lagrangian approach, a given volume represents a portion of material in the body, so that an observer follows the material as it moves through space.

Simcenter STAR-CCM+ employs both Lagrangian and Eulerian descriptions, whichever is most convenient for modeling a particular field of physics. For dispersed phases, Simcenter STAR-CCM+ offers a choice, as it implements both Eulerian and Lagrangian descriptions to describe similar phenomena.

Most of the physics represented in Simcenter STAR-CCM+ stem from a core set of fundamental laws. These fundamental laws are presented here in differential form, for an infinitesimal control volume. Further sections of the theory guide develop these laws into the numerical solution techniques that Simcenter STAR-CCM+ employs.

Mechanics

Continuum mechanics studies the behavior of continua in response to mechanical forces. The fundamental laws that govern the mechanics of fluids and solids are the conservation of mass, linear momentum, angular momentum, and energy.

Conservation of Mass

The balance of mass through a control volume is expressed by the continuity equation:

$$\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho v)=0$$

(654)

where $\rho$ is the density, that is, the mass per unit volume, and $v$ is the continuum velocity.

Conservation of Linear Momentum

The time rate of change of linear momentum is equal to the resultant force acting on the continuum:

$$\frac{\partial \left(\rho v\right)}{\partial t}+\nabla \cdot \left(\rho v\otimes v\right)=\nabla \cdot \sigma +f_b$$

(655)

where $\otimes$ denotes the outer product, $f_b$ is the resultant of the body forces (such as gravity and centrifugal forces) per unit volume acting on the continuum, and $\sigma$ is the stress tensor. For a fluid, the stress tensor is often written as sum of normal stresses and shear stresses, $\sigma =-pI+T$ , where $p$ is the pressure and $T$ is the viscous stress tensor, giving:

$$\frac{\partial \left(\rho v\right)}{\partial t}+\nabla \cdot \left(\rho v\otimes v\right)=-\nabla \cdot \left(pI\right)+\nabla \cdot T+f_b$$

(656)

Conservation of Angular Momentum

Conservation of angular momentum requires that the stress tensor is symmetric:

$$\sigma ={\sigma }^T$$

(657)

Conservation of Energy

When the first law of thermodynamics is applied to the control volume, the conservation of energy can be written as:

$$\frac{\partial \left(\rho E\right)}{\partial t}+\nabla \cdot (\rho Ev)=f_b\cdot v+\nabla \cdot (v\cdot \sigma )-\nabla \cdot q+S_E$$

(658)

where $E$ is the total energy per unit mass, $q$ is the heat flux, and $S_E$ is an energy source per unit volume.

Electromagnetism

Electromagnetism studies the behavior of continua in response to electromagnetic fields. The fundamental laws that describe the electromagnetic behavior of a continuum are Maxwell's equations and the conservation of electric charge.

Maxwell's Equations

Maxwell's equations can be written as:

$$\begin{array}{c}\frac{\partial \mathbf{\text{B}}}{\partial t}+\nabla \times \mathbf{\text{E}}=0\end{array}$$

(659)

$$\begin{array}{c}\frac{\partial \mathbf{\text{D}}}{\partial t}-\nabla \times \mathbf{\text{H}}=-\mathbf{\text{J}}\end{array}$$

(660)

$$\begin{array}{c}\nabla \cdot \mathbf{\text{D}}=\rho \end{array}$$

(661)

$$\begin{array}{c}\nabla \cdot B=0\end{array}$$

(662)

where $\rho$ is the electric charge density, $\mathbf{\text{J}}$ is the electric current density, $\mathbf{\text{E}}$ is the electric field, $\mathbf{\text{D}}$ is the electric flux density, $\mathbf{\text{H}}$ is the magnetic field, and $\mathbf{\text{B}}$ is the magnetic flux density.

Conservation of Electric Charge

The conservation of charge within a control volume is given by the continuity equation:

$$\nabla \cdot \mathbf{\text{J}}+\frac{\partial \rho }{\partial t}=0$$

(663)

Simcenter STAR-CCM+ also models chemical reactions, radiation, and a variety of other phenomena. The mathematical models that describe these phenomena are discussed in specific sections.

Constitutive Laws

In most cases, the partial differential equations of a mathematical model are not a closed set, that is, the number of unknown quantities exceeds the number of equations. To provide closure, additional equations are added to the mathematical models. These additional equations, called constitutive laws, depend on the material under consideration.

Boundary Conditions

Discretization and Solution

Simcenter STAR-CCM+ uses discretization methods to convert the continuous system of equations to a set of discrete algebraic equations, which can be solved using numerical techniques.

Depending on the mathematical model, Simcenter STAR-CCM+ discretizes the continuous equations using either the Finite Volume or the Finite Element method.