Conservation Principles

Control Volume V

Figure 3.1: Definition of a Control Volume

3.1 Conservation Principles

In order to define the conservation equations a quantity that is widely used in Fluid Dynamics, the so-called Material Derivative, has to be determined first.

This will be of particular use in order to simplify the equations and use them in a computational context. The conservation principles that will be derived are the conservation of mass, momentum and energy. These three principles are enough to calculate the state of a fluid, and the equations derived are usually coupled with each other, although some simplifications can be made in some instances, where the number of dependent variables reduces and therefore it is easier to find a solution to the particular mathematical problem.

3.1.1 Material Derivative

The Material Derivative (also known as Substantial Derivative) of a variable is defined as the instantaneous change of the variable itself of a fluid element and it is calculated in the following way:

Dψ Dt = ∂ψ

∂t + u · ∇ψ (3.1)

Where ψ can be any tensor field and u is the velocity field. Focusing on the last equation, we can therefore conclude that the variable change rate in time following a fluid element (Dψ/Dt) is equivalent to the local time derivative (∂ψ/∂t) plus the convective derivative of the same variable (u · ∇ψ). This means

3.1. CONSERVATION PRINCIPLES

that the state of a fluid element is changing as the element moves in the domain because the flow could change in time in that point and also because the element is moving to another point where the fluid has different properties.

3.1.2 Continuity Equation

One of the most important principles in fluid dynamics is the conservation of mass.

This follows the chemical principle that mass is neither created nor destroyed in any chemical reaction. Following the above, the continuity equation can be defined. If we consider a fluid region, the conservation of mass tells us that the sum of the mass that comes into a closed domain, the mass that leaves the domain and the mass change inside the domain equals to zero. In a differential form, the equation can be written using the material derivative as:

Dρ

Dt + ρ∇ · u = 0 (3.2)

The Continuity Equation could be also written developing the Material Deriva-tive introduced before, and it assumes the following form:

∂ρ

∂t + ∇ · (ρu) = 0 (3.3)

If we assume that the flow is incompressible, that is a flow where we can expect that the density is constant both in space and time (ρ = const), the equation becomes much more straightforward:

∇ · u = 0 (3.4)

3.1.3 Momentum Equation

In the study of mechanics of solid objects, Sir Isaac Newton derived in 1686 the second law of motion which correlates the change in momentum of an object to the external force applied to it. In a mathematical for this assumes the following equation:

F = ma (3.5)

The last equation can be used in our application as it can be easily applied to fluids as well. In the case of a fluid, this equations states that the momentum

3.1. CONSERVATION PRINCIPLES

change of a fluid element varies accordingly to the resultant forces applied to that element. For a fluid element, the forces applied to it include volume forces (such as gravity) and surface forces (i.e. viscous and pressure forces).

If we assume an infinitesimal fluid element and all the forces acting on each surface of the element, we can write Newton’s equation for all the three Cartesian axes in the following form: Where p is the pressure, τ is the viscous stress tensor and f are the volume forces. These three scalar equations are usually called Navier-Stokes equations because the two scientists discovered them independently during the nineteenth century. The viscous stress tensor components are calculated in the following way: first coefficient of viscosity (usually simply called dynamic viscosity) and λ is the second coefficient of viscosity. The latter is typically assumed to be zero in most cases, but if assumed different from zero the most common approximation is:

λ = −2

3µ (3.10)

With these assumptions, and developing the material derivative on the left hand side, we obtain the following vectorial form of the Navier-Stokes Equations:

ρ ∂u

3.1. CONSERVATION PRINCIPLES

3.1.4 Energy Equation

The main principle behind the energy equation is that energy is conserved, mean-ing that the energy rate of change in a material particle is equal to the energy received by heat and work to the particle. The equation can be written in the following way:

ρDe

Dt + ρDK

Dt + ∇ · (up) = −∇ · q + ∇ · (τ ⊗ u) + ρφ (3.12) where e is the specific internal energy, K = |u|²/2 is the local kinetic energy, q represents the heat flux, τ is the mechanical stress tensor and φ the heat source from other processes than the ones mentioned. Developing the material derivatives the equation becomes:

∂ρe

∂t + ∇ · (ρue) +∂ρK

∂t + ∇ · (ρuK) + ∇ · (up) = −∇ · q + ∇ · (τ ⊗ u) + ρφ (3.13) Sometimes the equation can be expressed in terms of enthalpy, assuming a similar expression to the one presented for the internal energy.

3.1.5 Conservation of Scalar Quantities

In several applications of fluid mechanics, in addition to the standard quantities (ρ, u, T ), a number of secondary scalar quantities can be present, such as chemical species or for example soot in the presence of a fire. These quantities evolve according to a transport equation (also called convection-diffusion equation for obvious reasons) where the change in time of the scalar is linked to convection and diffusion of the quantity itself and any other source term that could be present.

For a general scalar quantity φ it is possible to obtain a conservation equation as the ones derived before.

Dρφ

Dt + ρφ∇ · u − ∇²(ρDφ) = q_φ (3.14) where D is the diffusion coefficient of the scalar φ and qφ is a source terms that represent the transport of φ by mechanisms other that convection or diffusion and any sources or sinks of the scalar. Developing the material derivative with its definition the equation becomes: