Introduction

Fluid dynamics is an engineering discipline that describes the behavior of fluids in motion, i.e., the mechanisms by which fluids flow from one place to another and interact with objects. Fluids can be either liquids or gases. Aerodynamics, the study of airflow, is of particular interest to aerospace engineers, particularly in the context of wings, aircraft, and other bodies. In general, gas dynamics (or gasdynamics) is the study of the flow of gases, excluding air. Hydrodynamics is the study of water in motion, although the name hydrodynamics is also used to describe the behavior of different liquids.

The solution to fluid-dynamic problems typically involves predicting or measuring the fluid’s properties, such as velocity, pressure, density, temperature, and viscosity. These properties are called macroscopic fluid properties because they apply to a sizeable finite group of molecules of measurable dimensions, i.e., under the assumption of a continuum.^[1] Such fluid properties may be functions of the position (i.e., what is called spatial location) and, perhaps, of time (i.e., an unsteady flow). However, as a primer for understanding and predicting aerodynamic flows, it is necessary first to define the types of fluid flows encountered in engineering practice. The general categories are:

• • • • • External versus internal flows.
        • One-dimensional versus two-dimensional and three-dimensional flows.
        • Steady versus unsteady flows.
        • Laminar versus turbulent flows.
        • Viscous versus inviscid flows.
        • Incompressible versus compressible flows.

Laminar & Turbulent Flows

Fluid flows can be classified into two primary types: laminar and turbulent. There is no macroscopic mixing^[2] between fluid layers in a laminar flow; they move smoothly in layers or laminas. Laminar flows, however, transition naturally over time (or downstream distance) and become turbulent; such behavior is inevitable and predictable, at least in naturally occurring flow environments. In a turbulent flow, the fluid layers mix vigorously, and viscosity generates internal shear stresses that lead to the formation of spinning fluid elements and the development of eddies and turbulence, as shown in the flow-visualization image below.

Turbulent flows are the most common types of 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.

Reynolds’ Experiment

Consider now a classic flow experiment^[3] to examine the difference between laminar and turbulent flows, an investigation first conducted by Osborne Reynolds, where a dye was ejected into a pipe flow inside a tank of water, as illustrated in the figure below. Reynolds found that the flow was smooth (laminar) when the dye was introduced at low speeds. However, after a short distance, the flow naturally began to mix between the fluid layers, and after a further distance, the flow became thoroughly mixed. This mixed flow is called a turbulent flow, characterized by random eddies of various sizes and intensities. Reynolds noted that the higher the flow velocity, the quicker the flow transitioned to a turbulent flow – see here for flow visualization of the process. In his original paper, he also determined that the process depended on the properties of the fluid, specifically its density and viscosity.

Turbulent flows are difficult to describe in terms of measurement or calculation, and statistical methods must be employed. A turbulent flow is often called stochastic or non-deterministic because of its random nature, but turbulence is not chaotic.^[4]. 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, and the pipe diameter. He obtained a non-dimensional parameter involving the flow velocity, , density, , dynamic viscosity, , and the pipe diameter, , now known as the Reynolds number. The Reynolds number is given the symbol , and the equation is

(1)  

In general, the Reynolds number is defined as

(2)  

where is a characteristic length scale for that flow. The characteristic length scale is not unique, and its definition depends on the problem.

The Reynolds number quantifies the relative importance of viscous effects in a fluid. For Reynolds numbers below 2,000, the flow is generally laminar. When the Reynolds number exceeds approximately 4,000-5,000, the flow transitions from laminar to fully turbulent. The transition from laminar to turbulent flow is not a fixed point but a process, and so it occurs in the range of Reynolds numbers between these values. External influences may affect the transition Reynolds number range; upstream turbulence or external pressure disturbances tend to lower the Reynolds number at which transition occurs.

Ideal (Inviscid) 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. Further qualifications for an ideal (or perfect) fluid include the absence of surface tension and, if a liquid, the absence of vaporization. In general, 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 certain conditions, fluid behavior approaches the ideal, making the analysis much easier than if the flow were considered viscous and turbulent. Such ideal flows are often referred to as potential flows, for which a large body of technical literature exists. The field of potential flows encompasses many examples that span a broad range of practical applications for students of fluid dynamics.

The figure below shows the diversity of flow patterns obtained for varying Reynolds numbers, which measure the relative strength of inertial to viscous effects. At low values of , the inertia of the flow is smaller, so the effects of viscosity are more 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 complicates problem-solving in fluids, it is often convenient to divide real fluid flows into regions or zones that 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. Near-wall flows over surfaces or around bodies are inevitably viscous-dominated as a consequence of the shear produced in the flow. The critical part of this decomposition 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 (computationally) than one for an entirely inviscid solution. However, it remains substantially less time-consuming than a full viscous simulation.

Single Versus Multi-Dimensional Flow

In fluid dynamics, the distinction between 1-dimensional (1-D), 2-dimensional (2-D), and 3-dimensional (3-D) flows is based on the number of spatial dimensions that significantly affect the flow characteristics, as shown in the figure below. 1-D flow assumes that flow parameters (e.g., velocity, pressure, density) vary only along one spatial dimension. The parameters, however, could also vary with time. This 1-D simplification facilitates analysis of flows in situations where changes in the other two dimensions are negligible or uniform. Two-dimensional or axisymmetric flows can often be reduced to 1-D models by spatially averaging over one dimension, thereby simplifying the flow model.

2-D flow assumes that flow parameters vary in two spatial dimensions, while the third dimension is either uniform or negligible. This situation is often applicable when the flow has a large aspect ratio, meaning one dimension is much larger than the others. Examples include flow over a surface in which boundary-layer development along the length and height of the surface is considered, whereas variations across the width of the surface are assumed negligible. In some problems, if the width is much larger than the depth, variations in flow properties can be considered in the vertical and horizontal directions, while ignoring changes in the third direction.

3-D flow accounts for variations across all three spatial dimensions. This is the most general and complex case, capturing the complete flow behavior. Examples include the airflow around an aircraft; the flow varies significantly in all three dimensions because of the complex geometry and varying aerodynamic forces. Turbulent flows, in general, are 3-D because the velocity, pressure, and other properties vary along the length, width, and depth of the flow field.

1-D and 2-D models simplify mathematical analysis and computational effort, but at the cost of reduced accuracy in some cases. The choice among 1-D, 2-D, and 3-D models depends on the flow problem and the desired type of flow information. In engineering, simplified models are often used for initial design and analysis, while more detailed 3-D simulations are employed for final verification. The choice of 1-D, 2-D, or 3-D models depends on the type of problem and the information desired from the analysis.

External Versus Internal Flows

The key differences between these two types of flows lie in their boundaries and the resulting flow behavior, as shown in the figure below. External flows have at least one unbounded side, which accounts for boundary layers and wake effects. In contrast, internal flows are confined by solid boundaries on all sides, with greater emphasis on pressure drops and flow distribution within the confined space.

Examples of external flows include airflow over an aircraft wing, water flow over a ship’s hull, and wind over a building. These flows typically involve the development of a viscous region at the surface (the boundary layer) and, in some cases, flow separation. The shape and orientation of the object strongly influence external flows.

Internal flows occur when a fluid moves through a confined space or duct, with boundaries on all sides. Examples of internal flows are air or liquid flows through a pipe or duct. Internal flows are characterized by interactions with the confining walls, which can affect the pressure drop and flow distribution. Key phenomena in internal flows include laminar or turbulent flow regimes and pressure losses.

Steady Versus Unsteady Flows

A flow with properties that do not change as a function of time is called a steady flow. A steady-state flow is a condition in which the macroscopic flow properties, such as the velocity and pressure at a point, do not change with 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 time, as shown in the figure below on the right. Mathematically, for steady flows, then

(3)  

where is any fluid 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 remain constant over time, while all statistically varying properties, such as the component of the fluctuating velocity, are constant and continuous over time.

One reason for distinguishing between steady and unsteady flows is that the former is often more tractable to analyze and predict. To this end, eliminating time from the governing equations of a fluid-flow problem often yields significant simplification of both the equations and the mathematical and/or numerical techniques required to solve them.

Reynolds Decomposition

The figure below shows the difference between a statistically steady mean 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.,

(4)  

Notice that and are functions of time and space, but the averages are a function of space only. This process is called a Reynolds decomposition. Reynolds decomposition can also be extended to three dimensions, in which each quantity can be decomposed separately into its mean and fluctuating components.

The mean values can be obtained by averaging the flow properties over a long period, say , so that for a given spatial location in the flow, say , then

(5)  

The length required to obtain the desired information depends on the flow. The general rule is that must be long enough for statistical convergence, which often means as long as practically possible within the times and constraints of any experiment. Formal convergence studies may also be necessary to ensure that the results fall within accepted statistical bounds.

Therefore, the turbulent part is

(6)  

from which the root mean square (rms) or “strength” of the turbulent flow can be defined as

(7)  

In practice, these averaged values are computed from discrete data (i.e., flow measurements made at discrete time intervals), so summations replace integrals. For a discretely sampled time-history of values, performing Reynolds decomposition gives the mean component as

(8)  

where is the number of contiguous measurements. To ensure validity, the value of must be sufficiently large that statistical convergence is obtained, typically needing thousands of samples.

Reduced Frequency

One important parameter used to help categorize whether a flow is steady or unsteady is the reduced frequency, given the symbols, , and defined as

(9)  

where is a characteristic physical frequency of the unsteady flow, is a characteristic length scale, and is a reference flow velocity. For a wing or airfoil, the reduced frequency is normally defined in terms of its semi-chord, i.e., , and the freestream velocity.

For = 0, the flow is steady. For 0 0.05, the flow can be considered quasi-steady; that is, unsteady effects are generally minor, and for some problems, they may be neglected completely. However, such bounds are subjective and not based on rigorous analysis. In quasi-steady flows, behavior can be evaluated by applying the principles of steady flow under instantaneous boundary conditions, meaning flow adjustments are assumed to occur instantaneously.

Flows with characteristic reduced frequencies of 0.05 and above are generally considered unsteady, so the flow physics begins to change, and quasi-steady assumptions become invalid. Problems with characteristic reduced frequencies of 0.2 or above are considered highly unsteady, and unsteady effects will begin to dominate the flow characteristics. Such issues are much more difficult to determine because local flow properties also depend on the previous time, i.e., on the temporal history of lift and other aerodynamic forces.

Compressible Versus Incompressible Flows

The term “compressibility” applied to a fluid means that the fluid can be compressed, thereby reducing its volume and bringing its molecules closer together. Gases are easily compressed because the molecules are relatively far apart, but liquids are mostly incompressible. The volume and density of a gas can be easily altered by adjusting its pressure or volume, thereby changing the density, e.g., by squeezing or compressing it. Flows of any given fluid can also be regarded as compressible or incompressible, or both. For example, while gas is fundamentally compressible, it is usually accepted that if the flow velocities remain below a Mach number limit (usually less than 0.3), then the flow of a gas can be considered incompressible.

Compressible Fluid

Consider an experiment with a gas in a sealed cylinder, as shown in the figure below. As the piston moves downward, it decreases the volume, compressing the gas. The molecules are now closer together; i.e., the gas’s density and pressure increase, and more molecules strike a smaller area of the cylinder. Because work is done on the gas to compress it, its temperature increases. Similarly, if the piston moves upward, the volume and pressure decrease, thereby reducing the gas density and temperature. 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 where the density is constant everywhere is called an incompressible flow.

Gases are always compressible because their molecules are relatively far apart, so there is plenty of space to squeeze them closer together. Therefore, the density of gases changes readily with even modest changes in temperature and pressure. Likewise, all gas flows are compressible to varying degrees. However, many gas-dynamic and aerodynamic problems, even with minor changes in pressure and density, can be modeled as incompressible without significant loss of accuracy. This distinction is important because solving problems involving compressibility effects is more complex than solving those classified as incompressible.

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

(10)  

where is called the bulk compression modulus of the particular fluid, as suggested in the figure below. The minus sign indicates that a decrease in volume accompanies an increased pressure. The bulk compression modulus is a material property that characterizes the compressibility of a gas, i.e., how readily a unit volume of gas changes when the pressure acting on it changes.

Liquids are difficult to compress because their molecules are closer together, yet they still have finite distances between them and remain very mobile. For most problems, liquids can be treated as incompressible. The physics of compressibility in liquids must be considered only when sound propagation effects need to be considered, e.g., in unsteady flow problems. For a liquid, the density is related to temperature as a coefficient of expansion, just as in a solid. The speed of sound in a fluid can be calculated from the bulk modulus and the fluid density using

(11)  

Under flow conditions that involve only slight changes in velocity or pressure, even the flow of 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. Again, in accordance with all incompressible flows, the pressure deviation must be small compared to the reference (baseline) pressure. However, the flow must always be treated as compressible for subsonic speeds above a certain threshold, especially in cases involving shock waves.

Compressible Flows

The Mach number is often used to measure the significance of compressibility effects in a flow. The study of compressible flows is relevant to problems associated with high-speed aircraft, jet engines, rocket engines, rockets, and spacecraft exiting and re-entering the atmosphere. The figure below schematically shows the flow about a wing’s cross-section; the freestream 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.

It should be clear why the Mach number can be used to measure compressibility effects in a flow. First, however, it is essential to note that the freestream Mach number alone is not a reliable indicator of whether the local flow can be considered incompressible. This is because, as the flow approaches and passes around a body, such as an airfoil, it is locally accelerated, and its local velocity (particularly near its leading edge or nose) increases. For example, suppose the freestream Mach number is 0.3. As the flow accelerates over the upper surface of an airfoil, the local Mach number may be as high as 0.5 or 0.6, or even higher, depending on the airfoil’s angle of attack.

Compressibility Threshold

The threshold for the onset of compressibility effects in gases can be established mathematically: it occurs when the local value of exceeds approximately 0.3. For gases undergoing adiabatic compression or expansion, the bulk modulus is given by

(12)  

where is the ratio of specific heats. Using Eq. 10, then for finite changes in pressure and density

(13)  

Assuming the increase in dynamic pressure, is equal to the decrease in static pressure, (from the Bernoulli equation), where is the pressure of a fluid element initially at rest, then

(14)  

using . The speed of sound is , so substituting into the previous equation gives

(15)  

If the density varies less than 5%, then the flow can usually be treated as incompressible, i.e.,

(16)  

and solving for gives . Therefore, to ensure incompressible behavior in practical applications, the flow Mach number must be less than about 0.3.

To verify this result, consider a fluid element initially at rest, which has density . If this flow is accelerated to some velocity and some Mach number , the density of the fluid can be assumed to change according to the isentropic relationship

(17)  

where for air, the result given by Eq. 17 being plotted in the figure below.

Notice that the variation of from unity is negligible at low Mach numbers, confirming that for local Mach numbers below a threshold of about 0.3, it is convenient to make an incompressible flow assumption in analysis. Notice that

(18)  

For low Mach numbers (), this latter result can be approximated as

(19)  

thereby confirming the result given previously in Eq. 15. However, this incompressible-flow assumption cannot be readily applied at higher Mach numbers without introducing significant errors in predicting flow properties. Therefore, in compressible flows, the thermodynamic equation of state must be used to relate flow properties.

The corresponding isentropic flow equations for pressure and temperature are also helpful to consider. For the pressure ratio, then

(20)  

and for the temperature

(21)  

Another critical difference between incompressible and compressible flows arises from temperature changes. In incompressible flow, temperature changes are generally small. However, in a compressible flow, significant temperature changes may occur.

Swirling & Vortex Flows

Vortex flows are characterized by swirling motion in which fluid rotates about a central axis and forms cylindrical structures. They can be found in many natural phenomena, such as hurricanes, tornadoes, whirlpools, and wingtip vortices from airplanes and other flight vehicles. The formation of vortices is complex and depends on multiple factors, including variations in fluid velocity, pressure gradients, and boundary conditions.

Aeronautical engineers are particularly interested in wing tip vortices, which are the vortices formed and trailed in the wake 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, the vortex typically trails from the tip of the winglet. Besides the drag caused by the wingtip vortices, they can also pose a hazard to following aircraft, which may get caught up in the wake and may experience a wake-induced upset.

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

A closer inspection of the flow-visualization image above shows that the vortex flow is predominantly laminar near the inner (core) region, characterized by a smooth flow field with no interactions between adjacent fluid layers. A more detailed study shows that any turbulence present within this region will be either relaminarized or suppressed; even substantial eddies cannot penetrate the vortex boundary. Moving radially outward from the darker void, this smooth region is followed by a transitional region with both smooth and turbulent flow, characterized by eddies of various 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 thoroughly turbulent.

The second image above is a velocity field measurement of the swirling vortex flow, with arrows indicating the velocity vectors. 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 first be seeded with tiny smoke particles, which are then injected into it. The lasers flash in quick succession (microseconds separate the beams), and digital cameras capture images of the motion of smoke particles, which, on first examination, appear as star fields. The images can then be processed numerically to determine the resulting velocity field.

These flow observations and measurements are critical for verifying and validating (called V & V) computational fluid dynamics (CFD) models. For example, the figure below shows CFD predictions of the wing tip vortex rollup using two turbulence models, called turbulence closure models. Although both solutions yield similar bulk-flow properties, they differ in detailed quantitative properties and turbulence levels. As to which results are “correct,” can only be reconciled with respect to detailed measurements made in experiments, such as in the wind tunnel. Even then, additional measurements may be needed to ensure that the CFD accurately predicts the correct physical behavior for the appropriate reasons.

Summary & Closure

The ability to classify fluid flows is the first step in understanding them and developing mathematical models to solve for these flows. Classifications of fluid flows include fully laminar or turbulent flow and/or steady or unsteady flow. The solution to laminar flows is relatively straightforward compared with turbulent flows and can be obtained using deterministic mathematical models. However, including the temporal dimension in the solution of fluid flows makes the analysis of such unsteady flows substantially more difficult.

Turbulent flows are generally the most difficult to understand and predict because they are unsteady and non-deterministic, so statistical methods must be used to describe them. Most practical problems in aerodynamics involve turbulent flows of one kind or another, which contain eddies of various scales. Accurately predicting turbulent flows remains challenging, but progress continues, and researchers are working to improve understanding of turbulence and its role in aerodynamics.