Governing Equations of Fluid Flows

Abstract

To calculate the aerodynamic forces and moments acting on a flying vehicle, it is imperative to describe the flow pattern around the vehicle. The flow pattern thus obtained depends on the geometry of the vehicle, its orientation with respect to the oncoming freestream, altitude, and flying speed. Often, these patterns are very complex which necessitates to carry out both the experimental and theoretical studies to characterize the resultant flow. Moreover, the theoretical studies may utilize simplifying approximations in order to obtain any solution at all, and they must be matched with experimental data. Therefore, we must understand the fundamental laws that govern the fluid flow phenomena so that we can relate the theoretical results with experimental observations.

Appendices

Summary

Fluid mechanics is the branch of science which studies the statics and dynamics of fluids, and thus, it is further classified into fluid statics and fluid dynamics. As the name suggests, the fluid statics studies the fluid at rest, whereas the fluid dynamics investigates the effect of forces on the fluid in motion. To analyze a moving fluid, the fundamental laws such as the conservation principles of mass, momentum, and energy are used. Moreover, in arriving at the basic equations of motion one can either define the motion of individual molecules or can predict the gross behavior of molecules in a control volume. However, defining the motion of each and every molecule in the fluid is a cumbersome task, and usually not preferred as one is usually interested in knowing the average properties of the flow in a selected control volume. The appropriate size of the chosen control volume is very important, as the number of molecules in this elemental volume must be sufficiently large to have a meaningful statistical description of the flow. That is, the elemental control volume must be a continuous media.

The scalar (dot) product of the vectors $\overrightarrow{\mathrm {A}}$ and ${\overrightarrow{\mathrm {B}}}$ are given as

$$\begin{aligned} \overrightarrow{\mathrm {A}}.\overrightarrow{\mathrm {B}}&= \mathrm {\left| \overrightarrow{\mathrm {A}}\right| \left| \overrightarrow{\mathrm {B}}\right| \cos \alpha } \end{aligned}$$

where $\mathrm {\left| \overrightarrow{\mathrm {A}}\right| }$ and $\mathrm {\left| \overrightarrow{\mathrm {B}}\right| }$, respectively, denote the magnitudes of vectors $\overrightarrow{\mathrm {A}}$ and ${\overrightarrow{\mathrm {B}}}$, and ${\alpha }$ is smaller of the included angle.

The cross product of the vectors $\overrightarrow{\mathrm {A}}$ and ${\overrightarrow{\mathrm {B}}}$ is defined as

$$\begin{aligned} {\overrightarrow{\mathrm {A}}\times \overrightarrow{\mathrm {B}}}&= {\left( \left| \overrightarrow{\mathrm {A}}\right| \left| \overrightarrow{\mathrm {B}}\right| \sin \alpha \right) \cdot {\hat{\mathrm{n}}}=\overrightarrow{\mathrm {C}}} \end{aligned}$$

where $\overrightarrow{\mathrm {C}}$ is perpendicular to the plane containing $\overrightarrow{\mathrm {A}}$ and ${\overrightarrow{\mathrm {B}}}$, and ${\hat{\mathrm{n}}}$ is the unit vector a direction which obeys “right-hand rule”. That is, if we rotate ${\overrightarrow{\mathrm {A}}}$ into ${\overrightarrow{\mathrm {B}}}$ by curling our fingers, then the right-hand thumb will point in the direction of ${\overrightarrow{\mathrm {C}}}$.

A scalar quantity given as a function of coordinate space and time is called scalar field. For example, pressure, density, and temperature are scalar fields, given in Cartesian space as follows:

$$\begin{aligned} \mathrm {p}&= \mathrm {p\left( x,y,z,t\right) }\\ \mathrm {\rho }&= \mathrm {\rho \left( x,y,z,t\right) }\\ \mathrm {T}&= \mathrm {T\left( x,y,z, t\right) } \end{aligned}$$

Similarly, a vector quantity given as a function of coordinate space and time is called a vector field. For example, velocity is a vector field,

$$\begin{aligned} {\overrightarrow{\mathrm {v}}}&= {\mathrm{v}_{\mathrm{x}}{\hat{\mathrm{i}}}+\mathrm{v}_{\mathrm{y}}{\hat{\mathrm{j}}}+\mathrm{v}_{\mathrm{z}}{\hat{\mathrm{k}}}} \end{aligned}$$

where

$$\begin{aligned} \mathrm {v_{x}}&= \mathrm {v_{x}\left( x,y,z,t\right) }\\ \mathrm {v_{y}}&= \mathrm {v_{y}\left( x,y,z,t\right) }\\ \mathrm {v_{z}}&= \mathrm {v_{z}\left( x,y,z, t\right) } \end{aligned}$$

For a pressure field in Cartesian space, $\mathrm {p=p\left( x,y, z\right) }$, the gradient of $\mathrm {p}$ at a given point $\mathrm {\left( x, y\right) }$ in space is defined as

$$\begin{aligned} \mathrm {\nabla p}&= {\frac{\partial \mathrm{p}}{\partial \mathrm{x}}{\hat{\mathrm{i}}}+\frac{\partial \mathrm{p}}{\partial \mathrm{y}}{\hat{\mathrm{j}}}+\frac{\partial \mathrm{p}}{\partial \mathrm{z}}{\hat{\mathrm{k}}}} \end{aligned}$$

It is a vector whenever its magnitude is the maximum rate of change of $\mathrm {p}$ per unit length of the coordinate space at the given point and its direction is that of the maximum rate of change of $\mathrm {p}$ at the given point.

The divergence (positive) of a vector field $\mathrm {\mathrm {\overrightarrow{\mathrm {V}}=\overrightarrow{\mathrm {V}}\left( x,y, z\right) }}$ is a scalar field $\mathrm {\left( \overrightarrow{\nabla }.\overrightarrow{\mathrm {V}}\right) }$, which measures how much $\mathrm {\overrightarrow{\mathrm {V}}}$ spreads out at each point or for a negative divergence, how much $\mathrm {\overrightarrow{\mathrm {V}}}$ converges to the point. That is, the divergence represents the volume density of the outward flux of a vector field from a small elemental volume around a given point. Mathematically, the divergence is the scalar product of the $\mathrm {\nabla }$ operator and the vector field on which it acts. In Cartesian space, if ${\overrightarrow{\mathrm {V}}=\mathrm{V}_{\mathrm{x}}{\hat{\mathrm{i}}}+\mathrm{V}_{\mathrm{y}}{\hat{\mathrm{k}}}+\mathrm{V}_{\mathrm{z}}{\hat{\mathrm{k}}}}$, then ${\nabla .\overrightarrow{\mathrm {V}}}$ is

$$\begin{aligned} \mathrm {\overrightarrow{\nabla }.\overrightarrow{\mathrm {V}}}&= \mathrm {\mathrm {\frac{\partial }{\partial x}V_{x}+\frac{\partial }{\partial y}V_{y}+\frac{\partial }{\partial z}V_{z}}} \end{aligned}$$

In Cartesian space, if the flow velocity is given as ${\overrightarrow{\mathrm {v}}=\mathrm{v}_{\mathrm{x}}{\hat{\mathrm{i}}}+\mathrm{v}_{\mathrm{y}}{\hat{\mathrm{k}}}+\mathrm{v}_{\mathrm{z}}{\hat{\mathrm{k}}}}$; the curl of the velocity vector $\mathrm {\left( \overrightarrow{\nabla }\times \overrightarrow{\mathrm {v}}\right) }$ will be given by

$$\begin{aligned} {\overrightarrow{\nabla }\times \overrightarrow{\mathrm {v}}}&= {\begin{vmatrix}{\hat{\mathrm{i}}}&{\hat{\mathrm{j}}}&{\hat{\mathrm{k}}}\\ \mathrm {\frac{\partial }{\partial x}}&\mathrm {\frac{\partial }{\partial y}}&\mathrm {\frac{\partial }{\partial z}}\\ \mathrm {v}_{x}&\mathrm {v}_{y}&\mathrm {v}_{z} \end{vmatrix}} \end{aligned}$$

The continuum hypothesis allows us to surmise that at any instant $\mathrm {t}$, there exists a fluid particle corresponding to every point in space occupied by the fluid. Suppose a property such as temperature $\mathrm {T}$ of the fluid is to be specified, it could be done in two different ways. In the first approach, the property is specified as a function of the position in space and time, i.e., $\mathrm {T=T\left( x,y,z, t\right) }$. This is termed as Eulerian or field description, which essentially specifies the temperature of that fluid particle which happens to be at the location $\mathrm {\left( x,y, z\right) }$ at the given time $\mathrm {t=t_{1}}$. At time $\mathrm {t=t_{2}}$, the temperature $\mathrm {T\left( x,y,z, t_{2}\right) }$ is the temperature, not of the same particle, but of a different particle—the one that happens to be at location $\mathrm {\left( x,y, z\right) }$ at $\mathrm {t=t_{2}}$. The other approach of specifying a property in a moving fluid consists of identifying the fluid particles with some labels, following them around, and specifying their properties as a function of time. Usually, the particles are labeled by the space point they occupied at some initial time $\mathrm {t_{0}}$. Thus, $\mathrm {T\left( x_{0}, y_{0}, z_{0}, t\right) }$ refers to the temperature at time $\mathrm {t}$ of a particle which was at location $\mathrm {\left( x_{0}, y_{0}, z_{0}\right) }$ at $\mathrm {t_{0}}$. This approach of identifying material points and following them along is termed as the Lagrangian or the particle or the material description.

For a given velocity field $\mathrm {\left( \overrightarrow{\mathrm {v}}\right) }$, the relation between the total or material or Lagrangian derivative of density $\mathrm {\left( \frac{D\mathrm {\rho }}{D\mathrm {t}}\right) }$ and the local derivative of density $\mathrm {\left( \frac{\partial \mathrm {\rho }}{\partial \mathrm {t}}\right) }$ is given by

$$\begin{aligned} \mathrm {\frac{D\rho }{Dt}}&= \mathrm {\frac{\partial \rho }{\partial t}+\left( \overrightarrow{\mathrm {v}}.\overrightarrow{\nabla }\right) \rho } \end{aligned}$$

From the engineering applications point of view, four basic or primary laws must be satisfied for a continuous media. They are as follows:

• 1. Conservation of mass (continuity equation),

• 2. Newton’s second law (momentum equation),

• 3. Conservation of energy (first law of thermodynamics), and

• 4. Increase of entropy principle (second law of thermodynamics).

In addition to these primary laws, there are numerous secondary (or auxiliary) laws, frequently called constitutive relations, that apply to particular type of medium (or flow processes). The equation of state for perfect gas, Newton’s law of viscosity, etc., are some common examples of subsidiary laws. Moreover, the Hooke’s law for elastic solids is also an auxiliary law, which is widely used in solid mechanics.

The fundamental laws of fluid motion are as pertinent to individual particles as to a group of particles. That is, the laws obtained for finite control volumes are equally applicable when the control volume dwindles to a point in the flow field. The method of analysis, where large control volumes are used to obtain aggregate forces or transfer rates, is termed as integral analysis. On the other hand, when the analysis is applied to individual points in the flow field, the resulting equations are differential equations and the method is termed as differential analysis.

In reality, all properties and flow characteristics are usually expressed as a function of three space coordinates and time. That is, in general, fluid flows are of three-dimensional in nature. A two-dimensional flow is distinguished by the condition that all properties and flow characteristics are functions of two space coordinates and time, and hence do not change along the third space coordinate direction. A one-dimensional flow is a further simplification, where all properties and flow characteristics are assumed to be expressible as function of one space coordinate and time.

The quantitative and qualitative information of fluid flow can be obtained through flow visualization, graphical representation, and mathematical analysis. However, the visual representation of flow fields is an important tool in modeling the flow phenomena. In general, there are four basic types of line patterns used to visualize the flow, namely, timelines, pathlines, streaklines, and streamlines.

The line joining a set of fluid particles at a given instant is known as timeline.

A streamline is an imaginary line in a fluid flow drawn in such a manner that it is everywhere tangent to the velocity vector at the particular instant in time at which the observation is made.

A pathline is defined as a line in the flow field describing the trajectory of a given fluid particle. It is essentially the path traversed by the fluid particle through the coordinate space over a period of time.

A streakline is defined as the instantaneous line of all the fluid elements that have passed through the point of injection at some earlier time.

For an element which is rectangular in shape when the motion has just started. If the flow velocity significantly varies across the extent of the fluid element the corners may not move in unison, and thus the element may rotate and in addition, its shape may become distorted. If the adjacent sides of the fluid element rotate equally and in the same direction, we have pure rotation; however, if the adjacent sides rotate equally but in opposite directions, it leads to pure shearing motion.

The fundamental governing equations for an incompressible flow are the continuity and momentum equations. For steady incompressible flow, the continuity equation in differential form is written as

$$\begin{aligned} \mathrm {\frac{\partial \rho }{\partial t}+\overrightarrow{\nabla }.\left( \rho \overrightarrow{\mathrm {v}}\right) }&= \mathrm {0} \end{aligned}$$

For steady and incompressible flows, the momentum equation (also known as Navier–Stokes equation) can be written as

$$\begin{aligned} \mathrm {\frac{\partial \overrightarrow{\mathrm {v}}}{\partial \mathrm {t}}+\left( \overrightarrow{\mathrm {v}}.\nabla \right) \overrightarrow{\mathrm {v}}}&= \mathrm {\overrightarrow{\mathrm {g}}-\frac{\overrightarrow{\nabla }p}{\rho }+\nu \nabla ^{2}\overrightarrow{\mathrm {v}}} \end{aligned}$$

Circulation $\mathrm {\left( \Gamma \right) }$ is defined as the line integral of a velocity field around a closed curve.

and circulation per unit area is known as vorticity $\mathrm {\left( \zeta \right) }$.

$$\begin{aligned} \mathrm {\zeta }&= \mathrm {\frac{\Gamma }{s}} \end{aligned}$$

In vector notations, the vorticity is defined as

$$\begin{aligned} \overrightarrow{\mathrm {\zeta }}&= \mathrm {\overrightarrow{\nabla }\times \overrightarrow{\mathrm {v}}} \end{aligned}$$

If $\mathrm {\zeta =0}$, the flow is called an irrotational flow.

According to Bernoulli’s principle, an increase in the flow speed simultaneously results with a decrease in pressure or a decrease in fluid’s potential energy. This theorem is named after Daniel Bernoulli (1700–1782), a Swiss mathematician and physicist, who published it in his book Hydrodynamica in 1738. The Bernoulli’s theorem states that in a fluid flow, the sum of the static pressure, dynamic pressure, and the hydrostatic pressure along a streamline remains invariant. For steady flow, the Bernoulli’s equation is written as

$$\begin{aligned} \mathrm {p+\frac{1}{2}\rho v^{2}+\rho gz}&= \mathrm {constant} \end{aligned}$$

For an incompressible, inviscid, and irrotational flow, the unsteady form of Bernoulli’s equation is

$$\begin{aligned} \mathrm {\rho \frac{\partial \phi }{\partial t}+\rho \left( \frac{1}{2}\overrightarrow{\nabla }\phi .\overrightarrow{\nabla }\phi \right) +\rho gz+p}&= \mathrm {f\left( t\right) } \end{aligned}$$

The speed of an object (aircraft) relative to the surrounding air mass is called the airspeed. The pitot-static probe is the commonly used instrument to measure airspeed in the laboratory and on the aircraft. However, there exist subtle differences in the requirements for the two applications. To measure airspeed the correct value of density should be used in calculations. This requirement is, although, feasible in the controlled laboratory environment where the density is either almost invariant or its variation can be easily accounted. However, to calculate the airspeed in actual flight applications, one should consider the variation of density with ambient atmospheric pressure. There are five common conventions of airspeeds which are used in aerospace applications:

• True airspeed (TAS),

• Indicated airspeed (IAS),

• Calibrated airspeed (CAS),

• Equivalent airspeed (EAS), and

• Ground speed (GS).

Reynolds transport theorem (RTT), also known as Leibniz–Reynolds transport theorem, is essentially a three-dimensional generalization of Leibniz integral rule. The conservation laws of mass, momentum, and energy adopted from classical mechanics and thermodynamics, where the system approach is mainly followed, whereas in aerodynamics, it is often more convenient to work with control volumes as it is difficult to identify and follow a system of fluid particles. Thus, it is essential to relate control system approach and the control volume approach for certain fluid and flow properties, which indeed is achieved through RTT. In mathematical form, it is expressed as

The nonconservation form of the energy equation for the viscous fluid flow is

$$ \mathrm {\rho \frac{Du}{Dt}+\overrightarrow{\mathrm {\nabla }}\left( \rho u\overrightarrow{\mathrm {v}}\right) =-p\overrightarrow{\mathrm {\nabla }}\overrightarrow{\mathrm {v}}+\overrightarrow{\mathrm {\nabla }}\left( k\overrightarrow{\mathrm {\nabla }}T\right) +\Phi +S_{u}} $$

where $\Phi$ is the viscous dissipation term which is defined as

$$\begin{aligned} \mathrm {\Phi }&= \mathrm {\frac{2}{3}\mu \left[ \left( \frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}\right) ^{2}+\left( \frac{\partial u}{\partial x}-\frac{\partial w}{\partial z}\right) ^{2}+\left( \frac{\partial w}{\partial z}-\frac{\partial v}{\partial y}\right) ^{2}\right] }\\&+ \mathrm {\mu \left[ \left( \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right) ^{2}+\left( \frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right) ^{2}+\left( \frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\right) ^{2}\right] } \end{aligned}$$

Exercises

1.1 Descriptive Type Questions

• 1. Express the Kinetic energy of the fluid for the two-dimensional motion in terms of stream function.

• 2. Express the governing equations for an ideal fluid in terms of streamline coordinates.

• 3. Consider an Eulerian flow field. In Cartesian coordinate space, let the velocity and density fields are given by ${\overrightarrow{\mathrm {v}}=3{\hat{\mathrm{i}}}+\mathrm{2x}{\hat{\mathrm{j}}}}$ and $\mathrm {\rho =4y^{2}}$, respectively. Calculate the rate of change of density in Lagrangian reference frame, i.e., following a moving fluid element.

• 4. The Lagrangian flow field description is given by

  $$\begin{aligned} \mathrm {x}&=\mathrm{x}_{0}\exp \mathrm{k}\left( \mathrm{t-t}_{0}\right) \\ \mathrm {y}&=\mathrm{y}_{0}\exp -\mathrm{k}\left( \mathrm{t-t}_{0}\right) \\ \mathrm {z}&=\mathrm{z}_{0} \end{aligned}$$

  where ${\mathrm{k}=0.2\,\mathrm{s}^{-1}}$ is a constant. At time ${\mathrm{t}=\mathrm{t}_{0}=0}$, if a particle is located at ${\mathrm{x}_{0}=1}\,{{\mathrm {m}}}$, ${\mathrm{y}_{0}=1}\,{{\mathrm {m}}}$, and ${\mathrm{z}_{0}=0}$, then calculate the position of the particle at $\mathrm {t=2\, s}$.

• 5. For a given flow field, if the velocity components are given by

  $$\begin{aligned} \mathrm {u}&=-\mathrm{A}\exp \left\{ -\mathrm{ky}\right\} \cos \mathrm{kx}\\ \mathrm {v}&=-\mathrm{A}\exp \left\{ -\mathrm{ky}\right\} \sin \mathrm{kx} \end{aligned}$$

  where $\mathrm {A}$ and $\mathrm {k}$ are the constants. Find the stream function for this flow. Check whether this flow is rotational or irrotational?

• 6. Derive the equation of static equilibrium for a large mass of fluid, where different parts are held together by means of gravitational attraction.

• 7. Draw the streamline for the fluid motion, given by ${\psi =\mathrm{xy}}$, where ${\mathrm{x}\ge 0}$ and ${\mathrm{y}\ge 0.}$

• 8. Find out the equations of streamlines for the velocity field ${\mathrm{u}=\frac{\mathrm{Ax}}{\left( \mathrm{x}^{2}+\mathrm{y}^{2}\right) }}$ and ${\mathrm{v}=\frac{\mathrm{Ay}}{\left( \mathrm{x}^{2}+\mathrm{y}^{2}\right) }}$, where $\mathrm {A}$ is a constant.

• 9. Find the stream function for the potential function ${\phi =\frac{1}{2}\ln \left[ \frac{\left( \mathrm{x+a}\right) ^{2}+y^{2}}{\left( \mathrm{x-a}\right) ^{2}+\mathrm{y}^{2}}\right] }$.

• 10. A rocket is moving vertically upward against the gravitational force of the earth. If the rocket motor is operating under steady flow conditions, so that the pressure in the combustion chamber and the exhaust gas velocity at the nozzle exit do not change with respect to time. Calculate the forward velocity of the rocket at any instant $\mathrm {t}$ in a fixed reference frame. Neglect the frictional effects caused by the earth’s atmosphere.

1.2 Multiple Choice Questions

• 1. Consider the following statements:

  (I) The line joining a set of fluid particles at a given instant is known as timeline.

  (II) A streamline is an imaginary line in flow field drawn in such a manner that it is everywhere tangent to the velocity vector.

  (III) A streakline is defined as the instantaneous line of all the fluid elements that have passed through the point of injection at some earlier time.

  (IV) A pathline is defined as the trajectory of a given fluid particle.

  Which of the above statements is/are FALSE?

  • (a) I only

  • (b) I and III

  • (c) II and IV

  • (d) None

• 2. Let ${\rho _{\mathrm{a}}}$ is density of air at an altitude and ${\rho _{\mathrm{a},\text {SL}}}$ is the air density at standard sea level and ${\sigma =\frac{\rho _{\mathrm{a}}}{\rho _{\mathrm{a},\text {SL}}}}$. Which of the following express the relationship between equivalent airspeed and true airspeed?

  • (a) ${{\mathrm{v}_{\text {equivalent}}} = \mathrm{v}_{\text {true}}\times \sigma }$

  • (b) ${\mathrm{v}_{\text {true}} = {{\mathrm{v}_{\text {equivalent}}}\times \sqrt{\sigma }}}$

  • (c) ${{\mathrm{v}_{\text {equivalent}}} = \mathrm{v}_{\text {true}}\times \sqrt{\sigma }}$

  • (d) ${\mathrm{v}_{\text {true}} = {{\mathrm{v}_{\text {equivalent}}}\times \sigma }^{2}}$

• 3. For high-speed aircraft, the indicated airspeed has to be corrected for compressibility. The instrument, used to achieve it, is

  • (a) Altimeter

  • (b) Mach meter

  • (c) Gyroscope

  • (d) Pitot probe

• 4. If the velocity components in $\mathrm {x}$- and $\mathrm {y}$-directions are $\mathrm {u=2x+3y}$ and ${\mathrm{v}=6\mathrm{x}+2\mathrm{y}}$, respectively. The magnitude of vorticity will be

  • (a) 0

  • (b) 3

  • (c) 4

  • (d) 9

• 5. Consider the motion of two circular cylinders of different diameters in still air. If the velocities of these cylinders are inversely proportional to their radii, then the drag experienced by them will be

  • (a) same

  • (b) more for the cylinder with larger diameter

  • (c) less for the cylinder with larger diameter

  • (d) more for the cylinder with smaller diameter

• 6. Consider the following statements with respect to the continuity equation.

  (I) It is based on the postulation of the conservation of mass.

  (II) It is not applicable to compressible fluids.

  (III) It can only be used for the analysis of conserved quantity.

  Which of the above is/are correct?

  • (a) I only

  • (b) I and II

  • (c) I and III

  • (d) II and III

• 7. The smoke rings exiting the chimney are an example of

  • (a) Streamline

  • (b) Streakline

  • (c) Pathline

  • (d) Streamtube

• 8. In a fluid flow, if the streamline pattern is invariant even after sometime then the flow is

  • (a) steady

  • (b) unsteady

  • (c) uniform

  • (d) nonuniform

• 9. Which of the following statement is true?

  • (a) Streamlines and pathlines are same in an incompressible flow.

  • (b) Streamlines and streaklines are same in any fully developed flow.

  • (c) Streaklines and pathlines are same in a uniform flow.

  • (d) Streamlines and streaklines are same in a steady flow.

• 10. In a steady flow field, the material derivative of velocity of any fluid particle

  • (a) may or may not be zero

  • (b) is always zero

  • (c) is never zero

  • (d) is independent of convective derivative of flow velocity

1.2.1 Keys

• 1. (d)

• 2. (c)

• 3. (b)

• 4. (b)

• 5. (a)

• 6. (c)

• 7. (b)

• 8. (a)

• 9. (d)

• 10. (a)