# 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 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 and/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 finite group of molecules of measurable dimensions. Such fluid properties may be functions of 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.

Objectives of this Lesson

• Appreciate the differences between laminar and turbulent flows.
• Be able to distinguish between a steady flow and an unsteady flow.
• Know why it is important to distinguish between predominantly incompressible and 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. A flow visualization image of a fluid flow stream undergoing a transition from a smooth laminar flow to a mixed turbulent flow. Flow is from left to right.

Consider now a classic flow experiment to examine the difference between laminar and turbulent flows, an experiment 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 layers of fluid, and then after some further distance, the flow became more fully mixed. This mixed flow is called a turbulent flow, and it is 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. A classic experiment used to show the fundamental difference between a smooth laminar flow and a mixing or turbulent flow.

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 referred to as a random process. Osborne Reynolds found that this type of behavior was related to the flow velocity and the type of fluid in the pipe, and more specifically, a non-dimensional parameter involving flow density, velocity, and viscosity that became 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 for a short time after the flow is first formed. The third type of flow 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.

# 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 to do than if the flow was considered to be 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 at different 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 vortices and turbulence. In the limiting case, when , then the inertia effects in the flow dominate over viscous effects, and the flow approaches an ideal flow, which is one free of vortices and turbulence. The further implications of the Reynolds number parameter on fluid flows are considered later. The flow patterns about about a body (in this case a circular cylinder) depend on the Reynolds number of the flow, which is a measure of the relative effects of inertia to viscous effects.

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. In problem-solving, engineers often use zonal methods to help simplify their analysis in that zone-specific solution methods can be used and the results coupled together, with a great saving in terms of net effort and cost.

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. Similarly, if the piston moves upward, the volume and pressure decrease, so the gas density 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. A simple experiment illustrating the effects of compressibility on a gas. Squeezing the gas by pushing down on the piston will increase the pressure and density of the gas. Because work is also done on the gas to compress it, then its temperature will also increase.

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 are still very mobile), and for most problems, liquids may be considered as 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.

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 flow about a body, in this case a wing section, is significantly affected by the Mach number, which is the ratio of the free-stream flow velocity to the speed of sound.

# 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.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion
• For a Boeing 787 at its cruise condition, is the flow over the wing likely to be incompressible or compressible, and why?
• Consider the aerodynamic flow of a car on the highway. Is the flow over the car going to be steady or unsteady? Laminar or turbulent? Might the nature of the flow depend on speed? Explain.
• A fighter airplane is performing acrobatic maneuvers. Will the flow over the wing be steady or unsteady? Laminar or turbulent?
• An engineer assumes that the flow through a propeller is steady and incompressible. Are these reasonable assumptions? Explain.
• Do some research and make a shortlist of aerospace-related flow problems that are most likely not described by a continuum model.

Other Useful Online Resources 