Equations of Fluid Motion

J. Gordon Leishman

21 Equations of Fluid Motion

Introduction

Solving problems in fluid dynamics and aerodynamics requires correctly setting up the appropriate mathematical models of the flow field. Deriving mathematical equations describing fluid dynamics and aerodynamic flows is relatively straightforward, as it is a systematic process well established in engineering practice. However, all practical problems will inevitably require some assumptions and approximations to the equations to obtain solutions; a common and valid assumption is that air behaves as an ideal gas. Other assumptions include two-dimensional, steady, inviscid, and incompressible flow. Part of the skill in solving problems in fluid dynamics and aerodynamics is understanding which reference frames and which sets or subsets of equations are required.

Learning Objectives

Understand how conservation principles are applied to solve fluid-dynamic and aerodynamic problems.
Appreciate the various types of flow models that can be used to solve fluid problems.
Understand the concepts of mass flux and mass flow.
Know how to set up a finite control volume model of a fluid flow.
Appreciate how the Reynolds Transport Equation (RTE) is derived and its uses.

Setting Up Flow Models

Setting up flow models in fluid dynamics and aerodynamics involves creating mathematical representations of fluid behavior to analyze and predict flow patterns, including streamlines, pressures, velocities, and related parameters. These flow models form the foundation for understanding and solving real-world engineering problems in aerospace and other disciplines.

Problem Definition: The first step is to clearly define the problem to be solved, which inevitably raises additional questions. Try to identify the type of fluid flow – is it incompressible or compressible, steady or unsteady, laminar or turbulent? What is the geometry of the fluid system, the boundary conditions, and the desired outcomes, e.g., flow rates, flow velocities, pressure distributions, etc.?
Governing Equations: Select the appropriate governing equations likely to describe fluid flow behavior. These typically include the continuity, momentum, and energy equations. If justified, additional equations, such as the equation of state, may be applied to specific problems.
Assumptions and Simplifications: Make any assumptions and simplifications that might reduce the complexity of the equations while ensuring that they remain relevant to the problem. Typical assumptions include neglecting specific forces (e.g., viscosity) or considering steady-state conditions.
Boundary Conditions: Specify appropriate boundary conditions at the system’s boundaries. These conditions can include prescribed velocities, pressures, temperatures, and any other relevant parameters. Boundary conditions will be crucial in determining the fluid’s behavior within and outside the system.
Solution: Solve the governing equations analytically, numerically, or by a combination of methods to obtain the required flow properties.
Post-Processing: Analyze the results obtained from the flow model. If necessary, generate plots, tables, and other visualizations to gain insights into flow behavior and to validate the model against experimental measurements.

Conservation Equations

The three fundamental conservation principles of mechanics must be applied to solve the fluid dynamic or aerodynamic problem, namely:

Conservation of mass, i.e., mass is neither created nor destroyed.
Conservation of momentum, i.e., a force acting on a mass equals its time rate of change of momentum.
Conservation of energy, i.e., energy is neither created nor destroyed and can only be converted from one form into another.

The resulting mathematical equations should then describe the fluid dynamic or aerodynamic behavior of the flow of interest, at least within the bounds of the stated assumptions and approximations. The solution to these equations can be obtained analytically, numerically, or both, providing the engineer with the desired results.

Flow Models

There are two basic approaches used in fluid dynamics and aerodynamics:

An integral or finite control volume approach in which the equations are developed as they apply to a finite control volume surrounding the problem.
The differential or infinitesimal fluid element approach, in which the relevant equations apply at every flow point.

In both approaches, the control volume or fluid element may be fixed in space, allowing the flow to pass through it, or it may move with the flow and contain the same set of fluid molecules. The former approach, in which the model is fixed in space, is known as an Eulerian model, as illustrated in Figure 1. The latter, in which the model moves with the flow, is called a Lagrangian model. Each of these modeling approaches has advantages and disadvantages when applied to solving specific aerodynamic problems. In most cases, there will be a preferred approach for each situation.

An Eulerian representation is a fixed control volume in space, with fluid elements flowing in and out of it. A convecting control volume carrying the same fluid elements is called a Lagrangian representation.

For example, on the one hand, an integral approach can be used to find total effects, such as the forces on a body in the flow, without necessarily solving for all the point properties. Knowing what the fluid is doing at every point in the flow may not be necessary, and an integrated approach may be more appropriate. On the other hand, a differential approach, as shown in Figure 2, would be required if the desired outcome were the local distributions of flow velocity and pressure at points within the flow and over the body surface.

A fixed element with fluid flowing through it represents an Eulerian approach, while a moving volume that follows the fluid represents a Lagrangian approach.

Likewise, a Lagrangian approach may be preferred over an Eulerian approach because it makes the problem description more manageable for modeling the physical problem, providing a mathematical description, and/or developing a solution methodology. Part of the skills needed for fluid dynamics and aerodynamics problem-solving (and engineering problem-solving, in general) is deciding which type of foundational model to apply to specific problems. Sometimes, such decisions may not be immediately apparent, even to an experienced engineer, and different approaches may need to be tried tentatively before a suitable approach is determined.

For example, it may be desired to predict the velocity and pressure distribution over the surface of an airfoil or wing, as shown in Figure 3. The question is then: What basic form of the aerodynamic model should be used? In this case, the answer is a differential model, in which point properties, such as flow velocity, streamlines, and pressures, can be determined. Integral forms of the equations are appropriate only when the overall or integrated aerodynamic effects are required. The total lift on the wing is an integral quantity because it arises from the impact of the pressure distribution when it is resolved and integrated over the wing’s surface.

Point-flow properties require a local flow model, i.e., a differential flow model; however, integrated quantities can be determined using a control-volume approach.

In practice, the integral approach is usually easier to learn and work with, at least from a mathematical and/or numerical perspective. The differential form of the equations is appropriate when distributive quantities, such as velocity and pressure distributions over the wing’s surface, are needed, as these are usually more computationally expensive to obtain. Again, the relative cost of obtaining a solution for the flow properties may need to be factored into the final choice of the model.

Finite Control Volume Approach

To introduce the conservation laws of fluid dynamics, it is convenient to focus on finite-control-volume or integral models, which are helpful because they relate the fluid’s global properties. The concern is with the fluid properties entering the control volume versus the changes to these properties that occur within it. However, in many other practical problems, fluid properties at a point in the flow may be required; therefore, using the differential (fluid-element) model and applying this approach to problem-solving is usually necessary.

In the finite control volume approach, a closed surface is drawn to enclose a specific volume of flow, as shown in Figure 4. The symbol $S$ defines the area of the closed surface that bounds the control volume containing a fluid of volume ${\cal{V}}$ . The control volume is abbreviated to “C.V.” (denoted by ${\cal{V}}$ in mathematics), and the control surface is abbreviated to “C.S.” and denoted by $S$ in mathematics. This control surface (and control volume) must be selected to enclose the region of the flow to which the governing conservation equations will be applied. In some cases, the required control volumes may cover only part of the domain when certain flow conditions are specified elsewhere, as is common in practice. Engineers must develop a problem-solving technique to determine the most suitable control surface or volume, enabling the application of governing equations and the correct calculation of flow properties.

An Eulerian flow model is a finite control volume (C.V.), fixed in space, with the fluid flowing in and out across its control surface (C.S.).

All fluid properties can and must be allowed to vary with spatial location (i.e., with respect to ${x}$ , ${y}$ , and ${z}$ ) and in time $t$ so that

(1) $\begin{eqnarray*} \varrho & = & \varrho ( x, y, z, t ) \\[6pt] \vec{V} & = & \vec{V} (x, y, z, t) \end{eqnarray*}$

As previously described, $dS$ is a small elemental area of the control surface, and the vector $\vec{{n}}$ is the unit normal vector. Because the product $\vec{n} \, dS$ appears in the resulting equations for the flow, the elemental unit normal vector area is defined as ${d\vec{S} = \vec{n} \, dS}$ . Remember that by convention, $\vec{{n}}$ , and so also $d\vec{S}$ , always point outward from the control volume perpendicular to the control surface. For example, if the surface is oriented perpendicular to the flow in the ${x}$ direction (i.e., in the ${y}$ – ${z}$ plane), then $\vec{n} = (1, 0, 0)$ and if the surface is oriented perpendicular to the ${z}$ direction (i.e., in the ${x}$ – ${y}$ plane) then $\vec{n} = (0, 0, 1)$ .

Notice: Be cautious not to confuse the symbol for velocity (a vector ${\vec{V}}$ or $\bf V$ with the symbol for volume $\cal{V}$ or a “curly V.” Sometimes the symbol ${\volume}$ is used rather than $\cal{V}$ , but the meaning (volume) is the same.

Mass Flow and Mass Flux

Before deriving the fundamental equations of fluid dynamics or aerodynamics, it is essential to examine a concept vital to all of them: mass flow. Consider a small, fully permeable surface of differential area $dS$ that is oriented at some angle in a flow, as shown in Figure 5.

The mass flow through an area $dS$ requires calculating the flow velocity component normal to the surface.

Let the area $dS$ be small enough so that the velocity of the flow is constant across it, i.e., in the spirit of differential calculus. Then, consider the orientation of the small surface to be defined in terms of a unit normal vector $\vec{{n}}$ . The normal unit vector $\vec{{n}}$ establishes the orientation of the surface, where $\vec{{n}}$ is perpendicular to the surface and points in the chosen normal direction. The mass flow $d\overbigdot{m}$ through the surface $dS$ per unit time (the mass flow rate) will be given by

(2) $\begin{equation*} d\overbigdot{m} = \varrho V_{n} dS = \varrho ( \vec{V} \bigcdot \vec{n} ) dS \end{equation*}$

where $V_n = \vec{V} \bigcdot \vec{n}$ is the signed component of the resultant flow velocity normal to the surface. Its sign depends on the chosen direction of the unit normal vector $\vec{n}$ . Remember that if $\vec{{V}}$ is the velocity of the flow through the surface, then the component of the resultant flow velocity normal (perpendicular) to the surface is given by the dot-product

(3) $\begin{equation*} V_n = \vec{V} \bigcdot \vec{n} \end{equation*}$

The concept is more effectively visualized in two dimensions, as illustrated in Figure 6. Notice that the elemental unit normal vector area is defined as $d\vec{S} = \vec{n} dS$ , so that

(4) $\begin{equation*} d\overbigdot{m} = \varrho ( \vec{V} \bigcdot \vec{n} ) dS = \varrho ( \vec{V} \bigcdot d\vec{S} ) \end{equation*}$

Normal velocity component shown in 2D; standard fluid equations use the elemental unit normal vector area.

In this two-dimensional case, then

(5) $\begin{equation*} \vec{V} = u \, \vec{i} + v \, \vec{j} \end{equation*}$

and

(6) $\begin{equation*} \vec{n} = n_x \, \vec{i} + n_y \, \vec{j} \end{equation*}$

so

(7) $\begin{equation*} V_n = \left( u \vec{i} + v \vec{j} \right) \bigcdot \left( n_x \vec{i} + n_y \vec{j} \right) = u \, n_x + v \, n_y \end{equation*}$

In general, the total mass flow rate, $\overbigdot{m}$ , over a surface $S$ , is given by

(8) $\begin{equation*} \overbigdot{m} = \int_S \varrho ( \vec{V} \bigcdot \vec{n} ) dS = \int_S \varrho ( \vec{V} \bigcdot d\vec{S} ) \end{equation*}$

Mass flow rate has dimensions ( $\rm M L^{-3}$ ) ( $\rm L T^{-1}$ ) ( $\rm L^2$ ) = $\rm M T^{-1}$ and so the units will be in kg s $^{-1}$ in SI units or slugs s $^{-1}$ in USC units.

The mass flux is defined as

(9) $\begin{equation*} \frac{d\overbigdot{m}}{dS} = \varrho V_{n} = \varrho ( \vec{V} \bigcdot \vec{n} ) \end{equation*}$

which has dimensions ( $\rm M L^{-3}$ ) ( $\rm L T^{-1}$ ) = $\rm M T ^{-1} L^{-2}$ and so units of kg s $^{-1}$ m $^{-2}$ or slugs s $^{-1}$ ft $^{-2}$ . The mass flux terms like $\varrho u,\varrho v,\varrho w$ , etc., frequently occur in fluid dynamic problem solving, so the meaning of these terms should be understood. The concepts of mass flux and the elemental unit normal vector area are also used to derive the governing equations for fluid dynamics and aerodynamic flows.

Momentum & Energy Flow Rates

The corresponding momentum and energy flow rates can also be derived. The momentum flow rate through the surface $dS$ (i.e., the convective transport of momentum) is

(10) $\begin{equation*} d\overbigdot{\vec{P}} = \varrho \, ( \vec{V} \bigcdot d\vec{S}) \, \vec{V} \end{equation*}$

Therefore, the total momentum flow rate, $\overbigdot{\vec{P}}$ , over a surface, $S$ , is given by

(11) $\begin{equation*} \overbigdot{\vec{P}} = \int_S \varrho \, ( \vec{V} \bigcdot d\vec{S} ) \, \vec{V} \end{equation*}$

which is a vector equation with three components in Cartesian space. Momentum flow rate has dimensions of ( $\rm M L^{-3}$ ) ( $\rm L T^{-1}$ ) ( $\rm L^2$ ) ( $\rm L T^{-1}$ ) = $\rm M L T^{-2}$ and so its units will be kg m s $^{-2}$ in SI units or slugs ft s $^{-2}$ in USC units.

The kinetic energy flow $d(\overbigdot{KE})$ through the surface $dS$ per unit time (the kinetic energy flow rate) will be

(12) $\begin{equation*} d(\overbigdot{KE}) = \frac{1}{2} \varrho \, ( \vec{V} \bigcdot d\vec{S} ) \, V^2 \end{equation*}$

Therefore, the total flow rate of kinetic energy, $\overbigdot{KE}$ , over a surface $S$ , is given by

(13) $\begin{equation*} \overbigdot{KE} = \frac{1}{2} \int_S \varrho \, ( \vec{V} \bigcdot d\vec{S} ) \, V^2 \end{equation*}$

Kinetic energy flow rate has dimensions ( $\rm M L^{-3}$ ) ( $\rm L T^{-1}$ ) ( $\rm L^2$ ) ( $\rm L^2 T^{-2}$ ) = $\rm M L^2 T^{-3}$ and so the units will be those of power, i.e., kg m $^2$ s $^{-3}$ or J s $^{-1}$ or watts (W) in SI units, and units of lb-ft s $^{-1}$ in USC units.

Extensive & Intensive Properties

The conservation laws involve the rates of change of extensive properties, which are proportional to the mass of fluid contained within the control volume. The three extensive properties, which are all transportable by the flow, are those previously considered, i.e., mass, momentum, and energy, so that

(14) $\begin{equation*} B = \left\{ \begin{array}{ll} \mbox{Mass:} \quad m = m \quad (\beta = 1) \\[6pt] \mbox{Momentum:} \quad m \vec{V} = m (\vec{V}) \\[6pt] \mbox{Energy:} \quad E = m (e) \end{array} \right. \end{equation*}$

where $e$ is called the specific energy, i.e., energy per unit mass. In the sense of an infinitesimal fluid volume $d{\cal{V}}$ , then the mass is $\varrho \, d{\cal{V}}$ .

The extensive properties, i.e., $m$ , $m \vec{V}$ , and $E$ , which depend on the extent of the system, are designated by the general symbol $B$ . The corresponding intensive properties, denoted by $1$ , $\vec{{V}}$ , and ${e}$ , are generally expressed as “per unit mass” and designated by the symbol $\beta$ . The extensive and intensive properties are related locally by

(15) $\begin{equation*} \beta = \frac{dB}{dm} \end{equation*}$

so that

(16) $\begin{equation*} B = \int_m \beta \, dm \end{equation*}$

Because the density of a fluid can change from point to point, it is always best to express the governing equations in terms of per unit mass. The intensive properties do not depend on the system’s mass or extent.

Reynolds Transport Theorem

The Reynolds Transport Theorem (RTT) yields a general equation known as the Reynolds Transport Equation (RTE). This equation enables the conversion of fluid-transport equations from Lagrangian to Eulerian reference frames, thereby providing a valuable problem-solving tool. Because the RTE connects governing equations across different reference systems, it is sometimes called a link equation.

Flow Model

The approach to deriving the RTE proceeds by defining a fluid control system (sys) and a control volume (C.V.), as shown in Figure 7. The system is a collection of fluid molecules of density $\varrho(x, y, z, t)$ in a Lagrangian frame of reference that sweeps into and out of the C.V. At some time, $t - \delta t$ , the system moves toward the C.V. At the time, $t$ , the system, and the C.V. are coincident, i.e., they occupy the same space. At some later time, $t + \delta t$ , the fluid system moves out of the C.V. Therefore, this means that some of the fluid moves out of the C.V., some of the fluid remains inside the C.V., and some fluid comes into the C.V. to replace the fluid and properties that have moved out. In deriving the RTT, the focus shifts to tracking and formally quantifying the fate of this fluid.

Derivation

Consider a small element of volume $d {\cal{V}}$ as shown in Figure 8. The density of the flow is $\varrho = \varrho ( x, y, z, t )$ , and the absolute flow velocity is $\vec{V} = \vec{V} (x, y, z, t)$ . If $\beta$ is one of the transportable intensive properties, i.e., $1$ , $\vec{V}$ , or $e$ for mass, momentum, or energy, respectively, then the corresponding extensive property inside the small volume is

(17) $\begin{equation*} dB_{\rm sys} = \beta \, \varrho \, d {\cal{V}} \end{equation*}$

The net value of the extensive property can be obtained by integrating over the control volume.

Integrating over the entire system volume then gives

(18) $\begin{equation*} B_{\rm sys} = \oiiint_{\rm sys} \beta \, \varrho \, d {\cal{V}} \end{equation*}$

Similarly, integrating over the C.V. gives

(19) $\begin{equation*} B_{{\cal{V}}} = \oiiint_{{\cal{V}}} \beta \, \varrho \, d {\cal{V}} \end{equation*}$

Notice that at time $t$ , then

(20) $\begin{equation*} B_{\rm sys}(t) = B_{{\cal{V}}}(t) \end{equation*}$

but at time $t +\delta t$ , the system and control volume no longer occupy the same region.

Let $B_{\rm in}$ be an extensive property of the fluid coming into the C.V. and $B_{\rm out}$ be the same extensive property of fluid coming out of the C.V., as shown in Figure 9. It will be apparent then that

(21) $\begin{equation*} B_{{\cal{V}}}(t + \delta t) = B_{\rm sys}(t + \delta t) + B_{\rm in}(t + \delta t) - B_{\rm out}(t + \delta t) \end{equation*}$

After rearrangement then

(22) $\begin{equation*} B_{{\cal{V}}}(t + \delta t) - B_{\rm sys}(t) = B_{\rm sys}(t + \delta t) - B_{\rm sys}(t) + B_{\rm in}(t + \delta t) - B_{\rm out}(t + \delta t) \end{equation*}$

Principle of the inward and outward flow of the extensive property from the control volume.

Recall that $B_{\rm sys}(t) = B_{{\cal{V}}}(t)$ , so

(23) $\begin{equation*} B_{{\cal{V}}}(t + \delta t) - B_{{\cal{V}}}(t) = B_{\rm sys}(t + \delta t) - B_{\rm sys}(t ) + B_{\rm in}(t + \delta t) - B_{\rm out}(t + \delta t) \end{equation*}$

and dividing by $\delta t$ gives

(24) $\begin{equation*} \frac{B_{{\cal{V}}}(t + \delta t) - B_{{\cal{V}}}(t)}{\delta t} = \frac{ B_{\rm sys}(t + \delta t) - B_{\rm sys}(t )}{\delta t} - \frac{B_{\rm out}(t + \delta t)}{\delta t} + \frac{B_{\rm in}(t + \delta t)}{\delta t} \end{equation*}$

The outcome here begins to look like a differential equation. Notice that the last two terms in Eq. 24 can also be expressed as

(25) $\begin{equation*} \frac{B_{\rm out}(t + \delta t)}{\delta t} = \frac{B_{\rm out}(t + \delta t) - B_{\rm out}(t)}{\delta t} \end{equation*}$

because $B_{\rm out}(t) = 0$ , and also

(26) $\begin{equation*} \frac{B_{\rm in}(t + \delta t)}{\delta t} = \frac{B_{\rm in}(t + \delta t) - B_{\rm in}(t)}{\delta t} \end{equation*}$

because $B_{\rm in}(t) = 0$ .

In the limit as $\delta t \rightarrow 0$ , then Eq. 24 becomes

(27) $\begin{equation*} \frac{d}{dt} \left( B_{{\cal{V}}} \right) = \frac{d}{dt} \left( B_{\rm sys} \right) - \frac{d}{dt} \left( B_{\rm out} \right) + \frac{d}{dt} \left( B_{\rm in} \right) \end{equation*}$

By rearrangement, then

(28) $\begin{equation*} { \frac{d}{dt} \left( B_{\rm sys} \right) = \frac{d}{dt} \left( B_{{\cal{V}}} \right) + \frac{d}{dt} \left( B_{\rm out} - B_{\rm in} \right) } \end{equation*}$

which is a differential equation. Substituting for $B_{\rm sys}$ from Eq. 18 and ${B_{{\cal{V}}}}$ from Eq. 19 gives

(29) $\begin{equation*} \\[4pt] \frac{d}{dt} \oiiint_{\rm sys} \beta \, \varrho \, d {\cal{V}} = \frac{d}{dt} \oiiint_{{\cal{V}}} \beta \, \varrho \, d {\cal{V}} + \frac{d}{dt} \left( B_{\rm out} - B_{\rm in} \right) \end{equation*}$

This previous equation connects fluid properties from a Lagrangian perspective, in which the fluid moves with the flow, to those in an Eulerian perspective, in which the properties are evaluated at fixed points in space within a control volume (C.V.). However, the C.V. need not be fixed in size; its shape and size may also change over time.

The last term in Eq. 29 now requires further attention. It is apparent from the adopted flow model that some fluid comes into and out of the C.V., so there must be some net relative velocity of the fluid out of the C.V. Consider a small area of the control surface, C.S., of area $dS$ , its orientation specified by the unit normal vector $\vec{{n}}$ , as shown in Figure 10. The relative velocity of the flow out of the C.V. over the C.S. will be

(30) $\begin{equation*} \vec{V}_{\rm rel} = \vec{V} - \vec{V}_{\rm C.S.} \end{equation*}$

and for a fixed control volume

(31) $\begin{equation*} \vec{V}_{\rm C.S.} = 0 \Rightarrow \vec{V}_{\rm rel} = \vec{V} \end{equation*}$

The “relative velocity” $\vec{V}_{\rm rel}$ refers to the velocity of the fluid relative to the moving boundary of the control volume, essentially the difference between the absolute fluid velocity and the local velocity of the control surface. While many problems in fluid dynamics and aerodynamics will have a fixed C.V. in space and time, it is still possible that the C.V. can be deforming, hence the existence of the component $\vec{V}_{\rm C.S.}$ .

Calculating the net mass flux requires consideration of the relative flow velocities of the C.V. and the system.

The volumetric flow rate $dQ$ over the surface will be the product of the component of flow velocity $V_{\rm rel}$ that is normal to the area $dS$ over which it flows, i.e.,

(32) $\begin{equation*} dQ = \left( \vec{V}_{\rm rel} \bigcdot \vec{n} \right) \, dS = \vec{V}_{\rm rel} \bigcdot d\vec{S} \end{equation*}$

Evaluating the dot (scalar) product then gives the component of the volume flow normal to the surface. The corresponding mass flow rate is then found by multiplying by the local flow density, i.e.,

(33) $\begin{equation*} d\overbigdot{m} = \varrho (\vec{V}_{\rm rel} \bigcdot d\vec{S} ) \end{equation*}$

In terms of intensive properties, then

(34) $\begin{equation*} d\overbigdot{B}_{\rm net} = \beta \, \varrho ( \vec{V}_{\rm rel} \bigcdot d\vec{S}) \end{equation*}$

Integrating over the control surface gives

(35) $\begin{equation*} \overbigdot{B}_{\rm net} = \oiint_{S} d\overbigdot{B}_{\rm net} \end{equation*}$

Therefore,

(36) $\begin{equation*} \frac{d}{dt} \left( B_{\rm out} - B_{\rm in} \right) = \frac{d}{dt} \left( B_{\rm net} \right) = \overbigdot{B}_{\rm net} = \oiint_{S} d\overbigdot{B}_{\rm net} = \oiint_{S} \beta \, \varrho ( \vec{V}_{\rm rel} \bigcdot d\vec{S} ) \end{equation*}$

Reynolds Transport Equation (RTE)

Finally, after updating the terms in the preceding derivation and organizing everything in proper order, then

(37) $\begin{equation*} \frac{d}{dt} \oiiint_{\rm sys} \beta \, \varrho \, d {\cal{V}} = \frac{d}{dt} \oiiint_{{\cal{V}}(t)} \beta \, \varrho \, d {\cal{V}} + \oiint_{S(t)} \beta \, \varrho ( \vec{V}_{\rm rel} \bigcdot d\vec{S} ) \end{equation*}$

which is called the Reynolds Transport Equation (RTE). Notice that the elemental mass, $dm$ , contained in the fluid volume $d {\cal{V}}$ is given by $dm = \varrho \, d {\cal{V}}$ .

In other words, Eq. 37 states that the time rate of change of an extensive flow property, $B$ , in the system (sys) is equal to the time rate of change of $B$ in the control volume (C.V.) plus the rate at which $B$ is leaving through the control surface, i.e.,

(38) $\begin{equation*} \underbrace{\frac{d}{dt} \oiiint_{\mathrm{sys}} \beta \, \varrho \, d\mathcal{V}}_{\begin{tabular}{c} \scriptsize Time rate of \\[-3pt] \scriptsize change of $B$ \\[-3pt] \scriptsize inside the system.\end{tabular}} = \underbrace{\frac{d}{dt} \oiiint_{\mathcal{V}(t)} \beta \, \varrho \, d\mathcal{V}}_{\begin{tabular}{c} \scriptsize Time rate \\[-3pt] \scriptsize of change of $B$ inside\\[-3pt] \scriptsize the control volume.\end{tabular}} + \underbrace{\oiint_{S(t)} \beta \, \varrho ( \vec{V}_{\mathrm{rel}} \bigcdot d\vec{S}) }_{\begin{tabular}{c} \scriptsize Rate at which $B$ is \\[-3pt]\scriptsize leaving through the \\[-3pt] \scriptsize control surface.\end{tabular}} \end{equation*}$

The ordinary total derivative $d/dt$ on the left-hand side of Eq. 37 denotes the rate of change of the integrated extensive property of the material system. For a fixed control volume, the first term on the right-hand side may be written using a partial derivative because the region of integration does not change with time. For a moving or deforming control volume, the total time derivative shown in Eq. 37 is required.

The RTE is valuable because it allows one to relate flows in Lagrangian and Eulerian reference systems. It also helps to derive the conservation laws of fluid dynamics and aerodynamics for mass, momentum, and energy. The RTE is often referred to as a “link” equation because it connects the governing equations in both reference systems.

What are the units of the Reynolds Transport Equation?

The Reynolds Transport Equation (RTE) relates to the time rate of change of an extensive flow property, $B$ . The three extensive properties are mass, momentum, and energy. Therefore, the units of the RTE depend on the property being considered. However, in each case, the units represent the property per unit time; for example, for mass, the units would be M T $^{-1}$ , and for momentum, the units of the RTE would be M L T $^{-2}$ , i.e., force units.

Summary & Closure

Setting up flow models involves creating mathematical representations of fluid behavior, which are necessary to analyze and predict the fluid’s flow path and related properties. Control volume approaches are fundamental concepts in fluid dynamics and are used to analyze and understand fluid behavior in various systems. A control volume is often chosen as a fixed region in space that encloses a specific volume of fluid and is used to study the transport of mass, momentum, and energy across its boundaries. The control volume can be of any shape or size, depending on the problem.

The differential or infinitesimal fluid element is another approach in which the relevant equations apply at every point in the flow. In both cases, the control volume or the fluid element may be fixed in space and allow the flow to pass through it, or it may move with the flow and contain the same set of fluid molecules. The Reynolds Transport Equation (RTE) is very important (i.e., a “star” equation) in fluid dynamics and aerodynamics. It can be used to connect the flow physics of Lagrangian and Eulerian flow models.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

How might the physical fluids problem be solved, and how might it impact the choice of flow models? Discuss.
How might flow models be used in interdisciplinary fields like biomedical engineering? What challenges arise when applying flow models across different disciplines?
Research the Internet for examples of applications in which flow models have significantly contributed to solving real-world fluid dynamics and/or aerodynamic problems.
A Pitot probe is used on an airplane to measure dynamic pressure. Is this a Lagrangian or an Eulerian measurement? Explain.
Put yourself in the position of a course instructor. What might be effective ways to teach students about setting up flow models? How might instructors better balance theoretical concepts with practical applications?
Consider specific flow scenarios or complex geometries in which establishing accurate flow models remains challenging. How might researchers tackle these challenges?

Additional Online Resources

Navigate here to watch a video from the National Science Foundation on types of flow models.
Watch this video to learn more about the Lagrangian and Eulerian flow models.
This is an excellent video on the derivation of the Reynolds Transport Equation (RTE).
Video: Fluid Mechanics: Reynolds Transport Theorem.

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2026 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.