Introduction

Fluid dynamics is a discipline of engineering used to describe the behavior of moving fluids, i.e., how fluids flow from one place to another. Fluids can be either liquids or gases. Of particular interest to aerospace engineers is the field of aerodynamics, which is the study of the flow of air. In general, gasdynamics is considered to be the study of the flow of gases other than air. Hydrodynamics is the study of liquids in motion.

The solution to fluid dynamic problems usually involves the prediction or measurement of various properties of the fluid, such as its flow velocity, pressure, density, temperature, viscosity, etc. These properties are referred to as macroscopic fluid properties because the properties apply to a finite group of molecules of measurable dimensions. Such fluid properties may be functions of the position (i.e., what is usually called spatial location) and, perhaps, of time (i.e., an unsteady flow). However, as a primer to understanding and predicting aerodynamic flows, it is first necessary to define the types of fluid flows encountered in engineering practice, including laminar versus turbulent flows, steady versus unsteady flows, and incompressible versus compressible flows.

Laminar & Turbulent Flows

Fluid flows can be classified into two primary types, namely, laminar flows and turbulent flows. There is no mixing between fluid layers in a laminar flow, and they move smoothly in layers or laminas. Laminar flows, however, transition naturally after some time (or some downstream flow distance) and become turbulent; such behavior is inevitable, at least in a naturally occurring flow environment. In a turbulent flow, the fluid layers mix, the viscosity of the fluid creating internal shear stresses that result in the formation of spinning fluid elements and the creation of eddies and turbulence, as shown, for example, in the flow visualization image below.

Consider now a classic flow experiment to examine the difference between laminar and turbulent flows, an investigation first conducted by Osborne Reynolds, where a dye is ejected into a pipe flow, as illustrated in the figure below. When the dye was ejected at low flow speeds, Reynolds found that the dye exited in a smooth or laminar flow form. However, after a short distance, the flow naturally started to mix between the fluid layers, and then after some further distance, the flow became more fully mixed. This mixed flow is called a turbulent flow, filled with random turbulent eddies of various sizes and intensities. Reynolds noted that the higher the flow velocity, the quicker the flow transitioned to a turbulent flow. He also determined that the process depended on the properties of the fluid, specifically its density and viscosity.

Turbulent flows are not easy to describe from the perspective of measurement or calculation, and statistical methods must be used. A turbulent flow is often referred to as stochastic or non-deterministic because of its random nature. The formal concept of a stochastic process is classically called a random process. Osborne Reynolds found that this behavior was related to the flow velocity, the type of fluid in the pipe, as well as the diameter of the pipe. He proceeded to obtain a non-dimensional parameter involving the flow velocity, its density and viscosity, and the diameter of the pipe, which is now known as the Reynolds number.

Turbulent flows are the most common flows encountered in practice, but laminar flows can exist under some specific circumstances, at least briefly after the flow is first formed. The third flow type can be classified as a transitional flow in that this flow is neither completely laminar nor thoroughly turbulent. Laminar flows will “transition” through this state before becoming fully turbulent, which is a process, not a sudden event, hence the name.

Listening to a boundary layer?

How can you listen to a boundary layer? Well, it turns out that is easier than you think! It is necessary to have some pressure taps in the surface, with a connection to a piece of rubber hose – in the wind tunnel we usually use a special hose called “Tygon tubing.” The other end you can plug into your ear, much like that of a stethoscope. You will soon be able to hear the differences!

You will hear very little for a smooth laminar flow over the surface, except for a light swishing noise. For a fully turbulent flow, you will hear much more of a rumbling noise because of the pressure disturbances caused by turbulence. Finally, you will hear bursts of swishing and rumbling for a transitional flow. Fascinating!

Ideal Flow

A fluid that is not subjected to the action of viscosity or compressibility and does not flow in a turbulent manner is called an ideal or inviscid fluid. An ideal fluid flow has no internal dissipation of energy or other losses associated with the effects of viscosity and turbulence. In reality, no fluid can ever really be entirely ideal. However, under some circumstances, fluid behavior approaches the ideal. In this case, it can be considered as such for analyses, which is much easier than if the flow was considered viscous and turbulent.

The Reynolds number, which is given the symbol , is defined as

(1)  

where is the density of the fluid, is the velocity of the flow, is a characteristic length scale for that flow, and is the coefficient of dynamic viscosity for that fluid. The Reynolds number is used as a measure of the relative significance of viscous effects in a fluid.

For example, the figure below shows the diversity of different flow patterns obtained to varying values of the Reynolds number. At low values of , then the effects of viscosity are important. In this case, the flow experiences flow separation and the formation of swirling flows called vortices and turbulence. In the limiting case, when , the inertia effects in the flow dominate over viscous effects, and the flow approaches an ideal flow, free of vortices and turbulence. The further implications of the Reynolds number parameter on fluid flows are considered later.

Because turbulence makes problem-solving much more complex, it is often convenient to divide real fluid flows into different flow regions or zones, which may be considered either predominantly ideal and inviscid or predominantly viscous and turbulent. Such a division or zonal decomposition is often used to analyze aerodynamic problems, primarily when the regions with a turbulent flow are confined to small parts of the entire flow field, as shown in the figure below. The critical part of the approach is how the regions (zones) are coupled computationally, i.e., how information is passed between the zones. A flow simulation with a zonal method costs more than one for an entirely inviscid solution. However, it is still generally much less than a full viscous simulation.

Steady Versus Unsteady Flows

A flow that does not change as a function of time is called a steady flow. A steady-state flow refers to the condition where the macroscopic flow properties, such as the velocity and pressure at a point, do not change with respect to time, as shown in the figure below on the left. However, in a time-dependent flow, also known as an unsteady flow, the flow properties at a point will change with respect to time, as shown in the figure below on the right. Mathematically, for steady flows, then

(2)  

where is any property such as pressure, temperature, velocity, density, etc.

Unsteady flow phenomena are encountered in many engineering applications. Examples include the flows in turbomachinery and piston engines, helicopter aerodynamics, and aeroacoustics. A turbulent flow is an unsteady flow, by definition. However, a turbulent flow can be statistically steady. This definition means that the average flow velocity and other quantities are constant with respect to time, and all the statistically varying properties, such as the component of the fluctuating velocity, are constant with respect to time.

The figure below shows the difference between a statistically steady turbulent flow and a statistically unsteady flow. A flow property can be decomposed into a mean or average part, , and a statistically mean fluctuating part, , i.e.,

(3)  

This latter process has a special name, which is called a Reynolds decomposition.

One reason it is helpful to distinguish between steady and unsteady flows is that the former is often more tractable to understand, as well as to predict. To this end, eliminating time from the solution of the equations that govern a fluid flow problem usually results in a significant simplification of the relevant governing equations as well as the mathematical and/or numerical techniques needed to solve these equations.

Compressible Versus Incompressible Flows

The term “compressibility” applied to a fluid means that a fluid can be compressed, squeezing the fluid and bringing the molecules closer together. Gases are easily compressed because the molecules are relatively far apart, but liquids are essentially incompressible. The volume and density of a gas can be easily changed by changing its pressure or by changing the volume, thereby changing the density, e.g., by squeezing it or otherwise compressing it.

In this regard, consider a simple experiment with a gas in a sealed cylinder, as shown in the figure below. As the piston moves downward and decreases the volume and the pressure, the gas is compressed, and the molecules are now closer together, i.e., the density of the gas increases, and more molecules impact over a smaller area of the cylinder. Because work is also done on the gas to compress it, then its temperature will also increase. Similarly, if the piston moves upward, the volume and pressure decrease, so the gas density decreases and its temperature also decreases. Therefore, it can be concluded that a gas flow in which the density, , varies in either space and/or time is called a compressible flow. In contrast, a flow in which the density is constant everywhere is called an incompressible flow.

All gas flows are compressible to a lesser or greater degree. However, many gasdynamic and aerodynamic problems can be modeled as incompressible without significant accuracy loss. This distinction is important because solving problems involving compressibility effects is generally much more complicated than those that can be classified as incompressible.

Gases are always compressible because they have molecules that are relatively far apart. Therefore, the density of gases changes readily with even modest changes in temperature and pressure. In a gas, the density is related to temperature and pressure by the ideal gas law, given by . The change in pressure in a fluid, , may be written as

(4)  

where is the bulk compression modulus of the particular fluid. The minus sign indicates that a decrease in volume accompanies an increase in pressure. The bulk compression modulus is a material property characterizing the compressibility of the gas, i.e., how easily a unit volume of a gas can be changed when changing the pressure acting upon it. Furthermore, the speed of sound can be calculated from the bulk modulus and the fluid density using

(5)  

Liquids, however, are difficult to compress because the molecules are closer together (but they are still very mobile), and for most problems, liquids may be considered as being incompressible. Only when sound propagation needs to be considered in liquids is it necessary to consider the physics of their compressibility. For a liquid, the density is related to temperature as a coefficient of expansion, just as in a solid.

Under flow conditions that involve only slight changes in velocity or pressure, even gases may be considered incompressible. This assumption is usually appropriate for low-speed flow or what might be called low subsonic flows, i.e., flows where the flow velocities are much less than the speed of sound. However, the flow must always be considered compressible for flows at higher subsonic speeds, especially those involving shock waves.

Mach Number Effects

The Mach number is often used to measure the significance of compressibility effects. The study of compressible flows is relevant to problems associated with high-speed aircraft, jet engines, rocket motors, rockets, spacecraft exiting and re-entering the atmosphere, etc. In the figure below, which schematically shows the flow about a wing’s cross-section, the free-stream Mach number varies from low subsonic to supersonic. It is apparent that the effects of the Mach number are significant, and the flow patterns about the wing change significantly.

The images below are schlieren flow visualization about a model of a fighter airplane in a supersonic wind tunnel.  Notice that schlieren images are inevitably circular in shape because the technique uses parabolic mirrors. Notice the build-up in the number and intensity of the almost normal shock waves as Mach 1 is approached. In supersonic flight, the Mach cone becomes increasingly swept back with increasing flight Mach numbers. The angle is used to denote the Mach angle, as given by

(6)  

which is the half apex angle of the Mach cone.

Therefore, the Mach angle is 90 at , and its value decreases quickly beyond Mach 1, as shown in the figure below. Notice that even by a Mach number of 2, the Mach cone is already swept back 60 to a half apex angle of . For hypersonic speeds, the Mach angle is so steep that it can touch the body’s surface, so a blunt or bluff body shape is often a better design solution to keep the shock waves away from the surface.

Swirling Vortex Flows

Vortex flows are characterized by swirling motion, where fluid rotates around a central axis and form a cylindrical shape. They can be found in many natural phenomena, such as hurricanes, tornadoes, and whirlpools. The formation of vortices is complicated and depends on many different factors, including changes in fluid velocity, pressure gradients, and the presence of boundaries, among others.

Aerospace engineers are particularly interested in wing tip vortices. The term refers to the vortices formed and trailed behind the wingtips of an aircraft as it flies. These vortices are created from the difference in pressure between the upper and lower surfaces of the wing, which creates a tightly swirling flow of air that rotates around the wing tip. Wing tip vortices can also be a significant source of drag on the aircraft, which is called “induced drag.” If the aircraft has a winglet, as shown in the photograph below, then the  vortex typically trails from the tip of the winglet.

Vortex flows can also be studied in the wind tunnel. In the image shown below on the left, a vortex flow is identified by using smoke particles injected into the flow and illuminated by a thin laser sheet. The center of the vortex can be identified by the darker circular region, which contains less smoke. Wherever the local velocities are high enough to cause centrifugal forces on the smoke particles, no matter how tiny, they will spiral radially outward. The particles will reach a radial equilibrium location only when the centrifugal and pressure forces are balanced.

A closer inspection shows that the vortex flow is predominantly laminar near the inner or core region, which is marked by a region of very smooth flow where there are no interactions between adjacent fluid layers. A more detailed study shows that any turbulence present inside this region will be either relaminarized or suppressed; even substantial eddies will not be able to penetrate this vortex boundary. Moving radially outward from the darker void, it can be seen that this smooth region is followed by a transitional region with regions of both smooth flow and turbulent flow with eddies of different sizes.  Further outside this transition zone is a more highly turbulent region. This multi-region vortex structure concept differs from the descriptions assumed by mathematical models in that the flow is neither completely laminar nor completely turbulent.

The second image shown above is a velocity field measurement of the swirling vortex flow, the velocity vectors being shown by the arrows. The measurements were made using the Particle Image Velocimetry (PIV) technique, which uses thin sheets of high energy laser light to illuminate the flow region of interrogation. The flow must be seeded first with tiny smoke particles. The lasers flash in quick succession (micro-seconds) and special digital cameras capture images of the movement of the smoke particles, which look a bit like star fields. The images can then be processed numerically to determine the particle movements and hence the velocity field.

Summary & Closure

The ability to classify fluid flows is the first step in understanding flows and developing mathematical models to solve for such fluid flows. Classifications of fluid flows include fully laminar or turbulent flow and/or steady or unsteady flow. The solution of laminar flows is relatively straightforward when compared to turbulent flows and can be solved by using deterministic mathematical models.  However, including the dimension of time into the solution of fluid flows makes analyzing such unsteady flows commensurately more difficult.

Turbulent flows are generally the most difficult to understand and predict because, by their very nature, turbulence is unsteady and non-deterministic, so statistical methods must describe it. Most practical problems in aerodynamics involve turbulent flows of one kind or another, which contain eddies of various scales. The accurate prediction of turbulent flows remains a challenge, but progress continues to be made in this area, and researchers are working to improve the understanding of turbulence and its role in aerodynamics.