Chapter 3

Fluid kinematics

Fluid kinematics describe the fluid motion without considering the forces that cause that motion. We will explore two main views of the flow: the Lagrangian view, which follows individual fluid particles, and the Eulerian view, which observes the flow at fixed points in space. Although the Eulerian (fixed-point) view is more commonly used in the theory and simulation of fluid flows, the Lagrangian (particle-following) view will be essential when deriving some of the fundamental equations, as well as for understanding where certain features of the flow come from. Both approaches are often used together in numerical simulations. Flows are typically simulated in the Eulerian framework on a fixed grid, and for many applications the flow is analyzed a posteriori and/or visualized in the Lagrangian framework. For example, picture a high-speed flow simulation around an aircraft that is modeled on a fixed grid, and particle-following trajectories drawn to visualize the turbulent wake behind the vessel. Another example is the Lagrangian evolution of an oil spill in the ocean or a volcanic plume in the atmosphere, derived from Eulerian simulation output.

We will also introduce some useful concepts to describe the flow, namely the velocity potential and the stream function. These two scalar quantities are complementary to the vector field of velocity and together provide a complete description of the flow.

Lagrangian and Eulerian derivatives of a fluid property

We will start by first drawing a distinction between the Lagrangian and Eulerian derivatives. Consider a 3-dimensional quantity $\varphi$ that varies in space and time such that $\varphi = \varphi(x, y, z, t)$ . This can be a scalar, a vector, or a tensor, however, to keep things simple, suppose $\varphi$ is a scalar field. Let’s find its rate of change. Since it depends on $x$ , $y$ , $z$ , and $t$ , the rate of change of $\varphi$ along each of these dimensions must be taken into account. So, the total change of $\varphi$ (let’s call it $\delta\varphi$ , where $\delta$ is a small but finite increment) over spatial and temporal increments $\delta x$ , $\delta y$ , $\delta z$ , and $\delta t$ , is the sum of changes along each of these dimensions:

\delta\varphi = \frac{\partial \varphi}{\partial x} \delta x + \frac{\partial \varphi}{\partial y} \delta y + \frac{\partial \varphi}{\partial z} \delta z + \frac{\partial \varphi}{\partial t} \delta t

Divide by $\delta t$ to obtain:

\frac{\delta\varphi}{\delta t} = \frac{\partial \varphi}{\partial x} \frac{\delta x}{\delta t} + \frac{\partial \varphi}{\partial y} \frac{\delta y}{\delta t} + \frac{\partial \varphi}{\partial z} \frac{\delta z}{\delta t} + \frac{\partial \varphi}{\partial t}

Recall the definition of $\nabla$ (Eq. 2.18) and let the finite increment $\delta t$ approach $dt$ (and likewise for $\delta x$ , $\delta y$ , and $\delta z$ ), to obtain:

\frac{d\varphi}{dt} = \frac{\partial \varphi}{\partial x} \frac{dx}{dt} + \frac{\partial \varphi}{\partial y} \frac{dy}{dt} + \frac{\partial \varphi}{\partial z} \frac{dz}{dt} + \frac{\partial \varphi}{\partial t}

The above is equivalent to applying the chain rule to $\varphi$ with respect to time and assuming that the spatial dimension variables are functions of time ( $\varphi = \varphi(x(t), y(t), z(t), t)$ ). Recognize that by stating the dependence of position on time, we are implicitly stating that we are following a fluid particle. Then, recognize that the velocity in each direction is the rate of change of the position in that direction:

\frac{d\varphi}{dt} = \frac{\partial \varphi}{\partial t} + u \frac{\partial \varphi}{\partial x} + v \frac{\partial \varphi}{\partial y} + w \frac{\partial \varphi}{\partial z}

which states that the total change of $\varphi$ is due to the local (at fixed point in space) change over time, and due to spatial variations of $\varphi$ as the fluid particle moves through them. Finally, recall the definition of $\nabla$ (Eq. 2.18) to obtain:

\frac{d\varphi}{dt} = \frac{\partial \varphi}{\partial t} + \mathbf{u} \cdot \nabla \varphi

The term $\frac{d\varphi}{dt}$ is called the total derivative of $\varphi$ . It is also called a Lagrangian derivative, or material derivative, since it follows the motion of a fluid particle. The term $\frac{\partial \varphi}{\partial t}$ is called the Eulerian derivative, or partial derivative of $\varphi$ with respect to time. The term $\mathbf{u} \cdot \nabla \varphi$ describes how $\varphi$ changes due to its spatial variation and the flow of the fluid.

Although the term $\mathbf{u} \cdot \nabla \varphi$ is the dot product of $\mathbf{u}$ and $\nabla \varphi$ , the Lagrangian derivative in Eq. 3.5 can be expressed as an operator:

\frac{d}{dt} = \frac{\partial}{\partial t} + (\mathbf{u} \cdot \nabla)

The parentheses on the right-hand side indicate that that term acts as an operator on a field. Like we stated for the operator $\nabla$ in the previous chapter, the total derivative operator $\frac{d}{dt}$ cannot stand on its own, but is instead applied to a field.

Lagrangian derivative of a volume

Consider a fluid parcel with a constant mass but whose volume may change over time and is $\int_V dV = V$ . The total rate of change of that volume as it moves with the fluid is equal to the surface integral of the velocity field $\mathbf{u}$ through the surface $S$ that is bounding the volume $V$ :

\frac{d}{dt}\int_V dV = \int_S \mathbf{u} \cdot d\mathbf{S}

Recall now the divergence theorem (Eq. 2.28) to obtain:

\frac{d}{dt}\int_V dV = \int_V \nabla \cdot \mathbf{u} dV

Now, for a volume parcel so small that $\int_V dV = \Delta V \to 0$ , the velocity divergence can be considered to be constant over the volume, and the integral can be replaced by the volume itself:

\frac{d\Delta V}{dt} = \Delta V \nabla \cdot \mathbf{u}

We can derive a similar expression for the rate of change of a fluid property per unit volume $q$ , such that $q \Delta V$ is the amount of that quantity in a fluid parcel with the volume $\Delta V$ .

\frac{d}{dt} (q \Delta V) = \Delta V \frac{dq}{dt} + q \frac{d\Delta V}{dt}

Recall the material derivative of $\Delta V$ from Eq. 3.9 to obtain:

\frac{d}{dt} (q \Delta V) = \Delta V \frac{dq}{dt} + q \Delta V \nabla \cdot \mathbf{u}

\frac{d}{dt} (q \Delta V) = \Delta V \left( \frac{dq}{dt} + q \nabla \cdot \mathbf{u} \right)

This was for a fluid property that is defined per unit volume. Let’s now do the same for some property $\varphi$ that is defined per unit mass, such that $\varphi \rho \Delta V$ is the amount of that quantity in the fluid parcel with the volume $\Delta V$ and density $\rho$ (and mass $\rho \Delta V$ ).

\frac{d}{dt} (\varphi \rho \Delta V) = \rho \Delta V \frac{d\varphi}{dt} + \varphi \frac{d(\rho \Delta V)}{dt}

However recall that our fluid parcel has constant mass, so $\frac{d(\rho \Delta V)}{dt} = 0$ . Our total derivative becomes:

\frac{d}{dt} (\varphi \rho \Delta V) = \rho \Delta V \frac{d\varphi}{dt}

The Lagrangian derivative of a volume will come in handy when we derive the continuity equation in the next chapter.

Velocity potential

Velocity potential is defined as a scalar field $\phi$ such that the velocity field $\mathbf{u}$ is the gradient of $\phi$ :

\mathbf{u} = \nabla \phi = \begin{bmatrix} \frac{\partial \phi}{\partial x} \\ \frac{\partial \phi}{\partial y} \\ \frac{\partial \phi}{\partial z} \end{bmatrix}

The concept of the velocity potential is useful in fluid mechanics because it is often easier to work with a scalar field than a vector field. We will revisit it later in Chapter Surface gravity waves when we derive the equations of surface gravity waves.

Summary

In this chapter, we covered:

Lagrangian (material) and Eulerian (field) derivatives; the former follows a fluid parcel of constant mass as it moves through the flow field, while the latter is the rate of change at a fixed point (or volume) in space;
The Lagrangian derivative of volume, as well as of a fluid property per unit volume and per unit mass.

We’ll use these concepts in the next chapter where we derive the equations of continuity and motion.