19 Continuity Equation

Introduction

Having laid down the fundamental forms of the flow models used in aerodynamic analyses, the governing equations can now be formulated to describe fluid dynamic or aerodynamic flows. The approach uses the three physical conservation principles: mass, momentum, and energy. In this first case, the applicable physical principle is that mass can neither be created nor destroyed. The resulting governing equation is called the continuity equation. It is a general equation valid for three-dimensional, unsteady flows and applies to all types of flows, e.g., compressible or incompressible, viscous or inviscid.

Learning Objectives

  • Know how to derive the most general form of the continuity equation in its control volume or integral form.
  • Learn how to use the continuity equation to solve simple flow problems.
  • Show how the continuity equation can be derived from the Reynolds Transport Equation.

Flow Model

As previously discussed, the flow model is a control volume that may either be fixed in space with the fluid moving through it (the most common application), which is called an Eulerian description of the flow, or the volume can move with the fluid such that the identical fluid particles are inside it, which is called a Lagrangian model. In either case, the physical conservation principles must be applied to the fluid inside the control volume and any fluid crossing its boundaries.

Consider a fixed finite control volume {\cal{V}} bounded by a surface of area S, as shown in the figure below. 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.” and the control surface to “C.S.” All properties are 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*}

A finite control volume, fixed in space, with the fluid flowing in and out across a control surface.

At any point on the control surface, the velocity is \vec{V}, which is given in terms of the Cartesian components as

(2)   \begin{equation*} \vec{V} = u \, \vec{i} + v \, \vec{j} + w \, \vec{k} \end{equation*}

At the same point, the unit normal vector area is d\vec{S}. Also, let d{\cal{V}} be an elemental fluid volume inside the total control volume.

Conservation of Mass

The fundamental principle of the conservation of mass requires that the net mass flow out of the control volume over surface S is equal to the time rate of decrease of mass inside the control volume {\cal {V}}. Now, that physical statement must be translated into mathematics.

Following the concept of mass flow and mass flux discussed previously, the elemental mass flow across area dS is \varrho \, \vec{V} \bigcdot d\vec{S}. Remember that by convention, d\vec{S} always points out of the control volume, so the value of \vec{V}\bigcdot d\vec{S} will be positive. Therefore, the total mass flow rate (i.e., the integral of the mass flow rate over the entire surface area) is

(3)   \begin{equation*} \oiint_S \varrho \, \vec{V} \bigcdot d\vec{S} \end{equation*}

which can be physically interpreted as a net outflow leaving the control volume. Notice that the double integral here means the summation over the surface S, i.e., an area integral.

The small fluid mass contained within the elemental volume inside the C.V. is \varrho \, d{\cal {V}}. Hence, the total mass inside the C.V. is

(4)   \begin{equation*} \oiiint_{\cal{V}} \varrho \, d {\cal{V}} \end{equation*}

where the triple integral means a volume integral. So, the time rate of decrease of mass inside the C.V. is

(5)   \begin{equation*} -\frac{\partial}{\partial t}\oiiint_{\cal{V}} \varrho \, d {\cal{V}} \end{equation*}

noting that the minus sign represents the mass decrease, i.e., what is leaving the C.V.

Because the principle of conservation of mass requires that the net mass flow out of the control volume be zero, then Eq. 3 must equal Eq. 5, i.e.,

(6)   \begin{equation*} \oiint_S \varrho \, \vec{V} \bigcdot d\vec{S} = -\frac{\partial}{\partial t}\oiiint_{\cal{V}} \varrho \, d {\cal{V}} \end{equation*}

or

(7)   \begin{equation*} \underbrace{\frac{\partial}{\partial t}\oiiint_{\cal{V}} \varrho \, d {\cal{V}}}_{\begin{tabular}{c} \scriptsize Time rate of \\[-3pt] \scriptsize change of mass \\[-3pt] \scriptsize inside C.V. \end{tabular}} + \underbrace{ \oiint_S \varrho \, \vec{V} \bigcdot d\vec{S}}_{\begin{tabular}{c} \scriptsize Net mass \\[-3pt] \scriptsize flow rate \\[-3pt] \scriptsize out of C.V. \end{tabular}} = 0 \end{equation*}

This latter equation is called the continuity equation for a fluid flow, in this case, in the integral or control volume form. It is a general equation valid for three-dimensional, unsteady flows and applies to all types of flows, e.g., compressible or incompressible, viscous or inviscid. In addition, it can be used to relate aerodynamic phenomena over a finite region of the flow, e.g., the properties of the flow as it comes into and leaves the specified control volume. The unknowns in the equation include the flow velocities and the flow density.

Simplifications of the Continuity Equation

Various reductions or simplifications of the continuity equation can be used to solve practical problems, and these forms also allow commensurate simplifications in the overall mathematics. For example, for a steady flow, nothing changes with respect to time and \partial/\partial t \equiv 0. This simplification means that the continuity equation reduces to

(8)   \begin{equation*} \oiint_S \varrho \, \vec{V} \bigcdot d\vec{S} = 0 \end{equation*}

In other words, the mass flow that comes into the control volume per unit of time then leaves the control volume simultaneously, i.e., no mass accumulates inside the control volume. If this assumption can be justified, eliminating time dependencies in aerodynamic problems is a significant and worthwhile simplification in most forms of practical analysis.

In the case of a steady flow, the mass flow into the control volume equals the mass flow out of the control volume.

Proceeding further by assuming that the flow is a steady (\partial/\partial t \equiv 0) single stream system and one-dimensional, e.g., a uniformly axisymmetric flow, then the continuity equation is

(9)   \begin{equation*} \oiint_S \varrho \left( \vec{V} \bigcdot d\vec{S} \right) = \sum_{S} \varrho \left( \vec{V} \bigcdot \vec{A} \right) = 0 \end{equation*}

where \vec{A} is the total vector area, which is positive when it points out of the C.V., as shown in the figure below. A much easier way to express this one-dimensional result for the continuity equation is in scalar form such that

(10)   \begin{equation*} \sum_{S} \varrho \, A \, V = \overbigdot{m} = \mbox{constant} \end{equation*}

which is a statement that the summation of the mass flow rates through the control volume is constant for a steady flow.

 

Single stream system to illustrate the principle of conservation of mass for a fluid.

For the situation shown in the figure above with a single inlet and outlet, then

(11)   \begin{equation*} \sum \varrho \, A \, V = - \varrho_1 \left( \vec{V}_1 \bigcdot \vec{A}_1 \right) + \varrho_2 \left( \vec{V}_2 \bigcdot \vec{A}_2 \right) = 0 \end{equation*}

noting the sign on the first term because \vec{V}_1 is in the opposite direction to \vec{A}_1. This can also be written in scalar form in terms of magnitudes as

(12)   \begin{equation*} -\varrho_1 \, V_1 \, A_1 + \varrho_2 \, V_2 \, A_2 = 0 \end{equation*}

or

(13)   \begin{equation*} \varrho_1 \, V_1 \, A_1 = \varrho_2 \, V_2 \, A_2 = \overbigdot{m} = \mbox{constant}. \end{equation*}

which is just a statement that \overbigdot{m}_1 = \overbigdot{m}_2, i.e., mass flow is conserved.

If the flow is incompressible (\varrho = constant), then the continuity equation becomes

(14)   \begin{equation*} \oiint_S \vec{V} \bigcdot d\vec{S} = \sum_{S}\vec{V} \bigcdot \vec{A} = 0 \end{equation*}

which leads to a further reduction in the complexity because of the elimination of \varrho as an unknown so that only the flow velocities need to be related, i.e.,

(15)   \begin{equation*} Q = \sum_{S} \vec{V} \bigcdot \vec{A} = \mbox{constant} \end{equation*}

where Q is the volume flow rate. In this case, then

(16)   \begin{equation*} - \varrho_1 \, V_1 \, A_1 + \varrho_2 \, V_2 \, A_2 = 0 \end{equation*}

or

(17)   \begin{equation*} V_1 \, A_1 = V_2 \, A_2 \end{equation*}

which means that Q_1 = Q_2, i.e., volume flow is conserved.

Finally, consider a reduction to a steady (but compressible), uniform, one-dimensional flow in the x direction. In this case, \vec{V} = u \vec{i} and so

(18)   \begin{equation*} \oint_S \varrho \, \left( u \vec{i} \bigcdot d\vec{S} \right) = \sum_{S} \varrho \, u \, A = \sum_{S} \varrho \, u \, l (1) = 0 \end{equation*}

or

(19)   \begin{equation*} \varrho_1 \, u_1 \, l_1 = \varrho_2 \, u_2 \, l_2 \end{equation*}

Notice that the areas A_1 and A_2 become areas per unit length, i.e., l_1 and l_2, respectively.

Flow Through a Converging/Diverging Duct

Consider the steady, uniform flow through a converging/diverging duct with a circular cross-section with inlet area A_1 and outlet area A_2, as shown in the figure below. This is a classic problem when learning fluid dynamics. The two areas are known, such as by measurement. If the flow is assumed to be steady and uniformly axisymmetric (i.e., one-dimensional), then determine the relevant form of the continuity equation to relate the flow conditions at the outlet to those at the inlet.

Flow model of a converging/diverging duct with a circular cross-section.

The first step in the solution is to define a coordinate system and the control surface/volume over which to apply the principle of conservation of mass. In this case, the decision on the control volume is relatively easy as the duct itself bounds the flow. There can be no mass flow over the walls of the duct, and this naturally applies no matter what the duct’s shape is.

The flow is steady, so \partial/\partial t \equiv 0, and no further justification is needed in this case. However, nothing is mentioned about whether the flow is compressible or incompressible. Because air is a gas, it must be assumed that the flow is compressible and that density must be retained as a variable. Furthermore, suppose the flow is uniformly axisymmetric. In that case, the flow velocity changes only in one direction, i.e., in the x direction based on the adopted coordinate system, another significant simplification toward the solution of this problem.

In light of the preceding assumptions, therefore, in this case, then

(20)   \begin{equation*} \oiint_{S} \varrho \, \vec{V} \bigcdot d\vec{S} = \iint_{1} \varrho \, \vec{V} \bigcdot d\vec{S} + \iint_{2} \varrho \, \vec{V} \bigcdot d\vec{S} + \iint_{\rm walls} \vec{V} \bigcdot d\vec{S} = 0 \end{equation*}

where the latter term is zero because there is no mass flow over the walls, i.e.,

(21)   \begin{equation*} \iint_{\rm walls} \varrho \, \vec{V} \bigcdot d\vec{S} = 0 \end{equation*}

The mass flow coming into the control volume through the left-hand side (face 1) is

(22)   \begin{equation*} \iint_{1} \varrho \, \vec{V} \bigcdot d\vec{S} = -\left( \varrho \, \vec{V} \bigcdot \vec{A} \right)_1 = -\varrho_1 A_1 V_1 \end{equation*}

the one-dimensional assumption being used and the minus sign on the first term indicating that the flow is in the opposite direction to d\vec{S}. Similarly, the flow coming out of the right-hand side (face 2) is then

(23)   \begin{equation*} \iint_{2} \varrho \, \vec{V} \bigcdot d\vec{S} = \left( \varrho \, \vec{V} \bigcdot \vec{A} \right)_2 = \varrho_2 A_2 V_2 \end{equation*}

which is positive in this case because the flow is now in the direction of d\vec{S}. Therefore, because the flow is steady, the principle of conservation of mass states that the net mass flow is zero, so what mass flow comes into the control volume per unit time must equal the net mass flow out of the control volume per unit time, i.e.,

(24)   \begin{equation*} -\varrho_1 A_1 V_1 + \varrho_2 A_2 V_2 = 0 \end{equation*}

or simply that

(25)   \begin{equation*} \varrho_1 A_1 V_1 = \varrho_2 A_2 V_2 = \overbigdot{m} = \mbox{constant} \end{equation*}

Rearranging the latter equation gives the outlet conditions

(26)   \begin{equation*} \varrho_2 V_2 = \left( \frac{A_1}{A_2} \right) \varrho_1 V_1 \end{equation*}

i.e., the mass fluxes are related by the area ratio A_1/A_2. If the flow was further assumed to be incompressible, then \varrho = constant, and so

(27)   \begin{equation*} V_2 = \left( \frac{A_1}{A_2} \right) V_1 \end{equation*}

Finally, this latter result must be examined to see if it reconciles expectations and makes sense. Engineers get into the habit of asking such questions in practical problem solving, i.e., based on the final equation(s), does (do) the result(s) make physical sense?

For example, if the outlet area were to be smaller than the inlet area (i.e., A_2 < A_1), then the expectation is that the flow velocity will increase as it flows into and out of the control volume, which it does according to the equations because A_1/A_2 > 1. Notice that while this particular problem may appear easy, and indeed it is in this case, it provides an excellent example of how the conservation laws, in the integral form, can be applied to a fluid dynamics or aerodynamics problem.

Worked Example #1 – Calculating flow velocities in a converging duct

Consider the steady flow of a particular gas through a horizontal, converging pipe with an inlet diameter d_1 of 0.22 m and an outlet diameter d_2 of 0.16 m. The density of the gas is known to change from \varrho_1 = 0.91 kg/m{^3} at the inlet to \varrho_2 = 0.83 kg/m{^3} at the outlet. If the inlet flow velocity of the gas V_1 is 5.1 m/s, what is its exit velocity V_2? Assume one-dimensional flow.

The general form of the continuity equation is

    \[ \frac{\partial}{\partial t}\oiiint_{{\cal{V}}} \varrho \, d{\cal{V}} + \oiint_S \varrho \, \vec{V} \bigcdot d\vec{S} = 0 \]

In this case, for steady, one-dimensional flow, then

(28)   \begin{equation*} -\varrho_1 \, A_1 \, V_1 + \varrho_2 \, A_2 \, V_2 = 0 \end{equation*}

or

    \[ \overbigdot{m} = \mbox{constant} = \varrho_1 V_1 A_1 = \varrho_2 V_2 A_2 \]

where the density must be retained as a variable. In terms of diameters, then

    \[ \varrho_1 V_1 d_1^2 = \varrho_2 V_2 d_2^2 \]

And solving for V_2 gives

    \[ V_2 = \frac{\varrho_1 V_1 d_1^2}{\varrho_2 d_2^2} \]

Substituting the numerical values gives

    \[ V_2 = \frac{0.91 \times 5.1 \times 0.22^2}{0.83 \times 0.16^2} = 10.57~\mbox{m s$^{-1}$} \]

Flow Through a Branched Duct

Consider the flow of hydraulic fluid through a branch circuit of a pipe, as shown below. The objective, again, is to use the principles of the conservation of mass to determine a relationship between the flow properties and the inlet and outlet conditions. The inlet and outlet areas of the pipe are assumed to be known. Remember that the first step in the analysis is to think about a sketch of the control volume and the control surface and annotate it appropriately. It will be assumed that the fluid is incompressible, i.e., \varrho = constant, steady, and one-dimensional; the one-dimensional assumption is that the flow velocities are constant over every cross-section.

The governing equation for continuity of flow, in this case, becomes

(29)   \begin{equation*} \oiint_{S} \varrho \, \vec{V} \bigcdot d\vec{S} = \oiint_{1} \varrho \, \vec{V} \bigcdot d\vec{S} + \oiint_{2} \varrho \, \vec{V} \bigcdot d\vec{S} + \oiint_{3} \varrho\vec{V} \bigcdot d\vec{S} + \oiint_{\rm walls} \varrho \, \vec{V} \bigcdot d\vec{S} = 0 \end{equation*}

where the mass flow over the solid walls would be zero, i.e.,

(30)   \begin{equation*} \oiint_{\rm walls} \varrho \, \vec{V} \bigcdot d\vec{S} = 0 \end{equation*}

If the flow velocities are constant over their respective areas (the one-dimensional assumption), then

(31)   \begin{equation*} -\varrho A_1 V_1 + \varrho A_2 V_2 + \varrho A_3 V_3 = 0 \end{equation*}

And if the density is constant (which it is for a liquid), then

(32)   \begin{equation*} -A_1 V_1 + A_2 V_2 + A_3 V_3 = 0 \end{equation*}

noting the negative sign on the first term and its significance. Another way of looking at this latter result for the branch flow is to write it as

(33)   \begin{equation*} A_1 V_1 = Q = A_2 V_2 + A_3 V_3 \end{equation*}

where Q is the volume flow rate. Of course, if there was no third exit, then the problem could be reduced to the one previously considered, and

(34)   \begin{equation*} A_1 V_1 = A_2 V_2 = Q = \mbox{constant} \end{equation*}

In general, considering the flow out of the junction as positive and the flow into the junction as negative, then for steady flow at any junction, the algebraic sum of all the mass flows must be zero, i.e.,

(35)   \begin{equation*} \sum_{i = 1}^{N} \varrho_i \, A_i \, V_i = 0 \end{equation*}

For an incompressible flow, i.e., a liquid, then volume is also conserved, i.e.,

(36)   \begin{equation*} \sum_{i = 1}^{N} A_i \, V_i = 0 \end{equation*}

Continuity Equation from the RTE

Recall that the Reynolds Transport Equation (RTE) can be expressed as

(37)   \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{\partial}{\partial t} \oiiint_{\mathcal{V}} \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} \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*}

In the case of mass, then \beta = 1, and for a fixed C.V., the mass of the system and the C.V. are the same, so

(38)   \begin{equation*} \frac{D}{Dt} \oiiint_{\mathrm{sys}} \beta \, \varrho \, d\mathcal{V} = 0 \end{equation*}

Therefore, the RTE becomes

(39)   \begin{equation*} 0 = \frac{\partial }{\partial t} \oiiint_{{\cal{V}}} (1) \varrho \, d {\cal{V}} + \oiint_{S} (1) \varrho ( \vec{V} \bigcdot d\vec{S} ) \end{equation*}

or

(40)   \begin{equation*} \frac{\partial }{\partial t} \oiiint_{{\cal{V}}} \varrho \, d {\cal{V}} + \oiint_{S} \varrho \vec{V} \bigcdot d\vec{S} = 0 \end{equation*}

which will be recognized as the continuity equation in its integral form.

Differential form of the continuity equation

The differential form of the continuity equation can also be derived from the RTE. Recall that using the divergence theorem, the RTE for a fixed control volume becomes

    \[ \oiiint_{{\cal{V}}} \bigg[ \frac{\partial (\beta \, \varrho)}{\partial t} + \beta \, \varrho \,\left( \nabla \bigcdot \vec{V} \right) \bigg] d{\cal{V}} = 0 \]

So, a differential form of the RTE becomes

    \[ \frac{\partial (\beta \, \varrho)}{\partial t} + \beta \, \varrho \,\left( \nabla \bigcdot \vec{V} \right) = 0 \]

where in this case, \beta = 1 and the mass of the system and the C.V. are the same. Therefore,

    \[ \frac{\partial \varrho}{\partial t} + \varrho \,\left( \nabla \bigcdot \vec{V} \right) = 0 \]

which will apply at every point in the flow. If the flow is incompressible, then \varrho = constant, so the continuity equation becomes

    \[ \nabla \bigcdot \vec{V} = 0 \]

or in terms of the scalar components when \vec{V} = (u, v, w), then

    \[ \nabla \bigcdot \vec{V} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \]

Therefore, this equation states that the conservation of mass requires that the divergence of the velocity field must be identically zero. If it is not, then the flow would be non-physical.

Summary & Closure

Applying the principle of the conservation of mass to fluids results in a governing “star” equation called the continuity equation. This equation applies to all fluids, viscous or inviscid, compressible or incompressible, steady or unsteady. In application, the continuity equation can be simplified from its most general form by making various assumptions as they may apply to the problem of interest. However, it should be remembered that all assumptions must be justified, which in some cases may be challenging to establish a priori, so caution should be used. The most common simplification is to write the continuity equation in one-dimensional form, i.e., in terms of fluid properties that change in only one direction. While the continuity equation can help solve certain simple classes of fluid flow problems, the solution of most real problems will generally need to invoke conservation principles of momentum and energy to elicit the required information.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

  • Think about a situation where the size of the control volume changes with time. What form of the continuity equation would be needed in this case?
  • Think of some fluid flow problems where a Lagrangian flow model might be preferable to solve the problem.
  • How might the continuity equation be applied to analyze blood flow in arteries and veins in the human circulatory system?
  • Explain how the continuity equation could be used to design and analyze water distribution systems, such as municipal water supply networks.
  • How does the continuity equation relate to streamlines in fluid flow? Provide an example to illustrate this relationship.

Additional Online Resources

  • A good video on some basics of the continuity equation.
  • Another video on the use of the continuity equation in fluid mechanics.
  • A simple application of the conservation of mass for a fire hose.

License

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Introduction to Aerospace Flight Vehicles Copyright © 2022, 2023, 2024 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/https://doi.org/10.15394/eaglepub.2022.1066.n16